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Exercícios de funções · Physics Playground
Exercícios
Matemática
Funções
93 problemas com soluções resolvidas, de Algebra and Trigonometry e Precalculus.
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Algebra and Trigonometry
3.1 Functions and Function Notation
Exemplo 1
Determining If Menu Price Lists Are Functions
The coffee shop menu, shown below, consists of items and their prices. ⓐ Is price a function of the item? ⓑ Is the item a function of the price?
Com figura
Exemplo 2
Determining If Class Grade Rules Are Functions
In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? Table 1…
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Exemplo 3
Using Function Notation for Days in a Month
Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.
Exemplo 4
Interpreting Function Notation
A function N=f( y) gives the number of police officers, N, in a town in year y. What does f( 2005)=300 represent?
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Exemplo 5
Identifying Tables that Represent Functions
Which table, Table 6, Table 7, or Table 8, represents a function (if any)? Input Output 2 1 5 3 8 6 Table 6 Input Output –3 5 0 1 4 5 Table 7 Input Output 1 0 5 2 5 4 Table 8
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Exemplo 6
Evaluating Functions at Specific Values
Evaluate f( x)=x^2 +3x−4 at: ⓐ 2 ⓑ a ⓒ a+h ⓓ Now evaluate (f( a+h)−f( a))/h
Exemplo 7
Evaluating Functions
Given the function h( p)=p^2 +2p, evaluate h( 4).
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Exemplo 8
Solving Functions
Given the function h( p)=p^2 +2p, solve for h( p)=3.
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Exemplo 9
Finding an Equation of a Function
Express the relationship 2n+6p=12 as a function p=f( n), if possible.
Exemplo 10
Expressing the Equation of a Circle as a Function
Does the equation x^2 +y^2 =1 represent a function with x as input and y as output? If so, express the relationship as a function y=f( x).
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Exemplo 11
Evaluating and Solving a Tabular Function
Using Table 11, ⓐ Evaluate g( 3). ⓑ Solve g( n)=6. n 1 2 3 4 5 g( n) 8 6 7 6 8 Table 11
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Exemplo 12
Reading Function Values from a Graph
Given the graph in Figure 4, ⓐ Evaluate f( 2). ⓑ Solve f( x)=4. Figure 4
Com figura
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Exemplo 13
Determining Whether a Relationship Is a One-to-One Function
Is the area of a circle a function of its radius? If yes, is the function one-to-one?
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Exemplo 14
Applying the Vertical Line Test
Which of the graphs in Figure 9 represent(s) a function y=f( x)? Figure 9
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Exemplo 15
Applying the Horizontal Line Test
Consider the functions shown in Figure 9(a) and Figure 9(b). Are either of the functions one-to-one?
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Algebra and Trigonometry
3.2 Domain and Range
Exemplo 1
Finding the Domain of a Function as a Set of Ordered Pairs
Find the domain of the following function: { ( 2,10),( 3,10),( 4,20),( 5,30),( 6,40) }.
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Exemplo 2
Finding the Domain of a Function
Find the domain of the function f(x)=x^2 −1.
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Exemplo 3
Finding the Domain of a Function Involving a Denominator
Find the domain of the function f(x)=(x+1)/(2−x).
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Exemplo 4
Finding the Domain of a Function with an Even Root
Find the domain of the function f(x)=√(7−x).
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Exemplo 5
Describing Sets on the Real-Number Line
Describe the intervals of values shown in Figure 6 using inequality notation, set-builder notation, and interval notation. Figure 6
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Exemplo 6
Finding Domain and Range from a Graph
Find the domain and range of the function f whose graph is shown in Figure 9. Figure 9
Com figura
Exemplo 7
Finding Domain and Range from a Graph of Oil Production
Find the domain and range of the function f whose graph is shown in Figure 11. Figure 11 (credit: modification of work by the U.S. Energy Information Administration)4…
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Exemplo 8
Finding the Domain and Range Using Toolkit Functions
Find the domain and range of f(x)=2x^3 −x.
Exemplo 9
Finding the Domain and Range
Find the domain and range of f(x)=2/(x+1).
Exemplo 10
Finding the Domain and Range
Find the domain and range of f(x)=2√(x+4).
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Exemplo 11
Writing a Piecewise Function
A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, n, to the cost, C.
Exemplo 12
Working with a Piecewise Function
A cell phone company uses the function below to determine the cost, C, in dollars for g gigabytes of data transfer. C(g)={ 25 if 0<g<2 25+10(g−2) if g≥2 Find the cost of using 1.5 gigabytes of data and the cost of…
Exemplo 13
Graphing a Piecewise Function
Sketch a graph of the function. f(x)={ x^2 if x≤1 3 if 1<x≤2 x if x>2
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Algebra and Trigonometry
3.3 Rates of Change and Behavior of Graphs
Exemplo 1
Computing an Average Rate of Change
Using the data in Table 1, find the average rate of change of the price of gasoline between 2007 and 2009.
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Exemplo 2
Computing Average Rate of Change from a Graph
Given the function g( t) shown in Figure 1, find the average rate of change on the interval [ −1,2 ]. Figure 1
Com figura
Exemplo 3
Computing Average Rate of Change from a Table
After picking up a friend who lives 10 miles away and leaving on a trip, Anna records her distance from home over time. The values are shown in Table 2. Find her average speed over the first 6 hours. t (hours) 0 1 2…
Exemplo 4
Computing Average Rate of Change for a Function Expressed as a Formula
Compute the average rate of change of f( x)=x^2 −1/x on the interval [2,4].
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Exemplo 5
Finding the Average Rate of Change of a Force
The electrostatic force F, measured in newtons, between two charged particles can be related to the distance between the particles d, in centimeters, by the formula F( d)=2/(d^2). Find the average rate of change of…
Exemplo 6
Finding an Average Rate of Change as an Expression
Find the average rate of change of g( t)=t^2 +3t+1 on the interval [0,a]. The answer will be an expression involving a in simplest form.
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Exemplo 7
Finding Increasing and Decreasing Intervals on a Graph
Given the function p( t) in Figure 6, identify the intervals on which the function appears to be increasing. Figure 6
Com figura
Exemplo 8
Finding Local Extrema from a Graph
Graph the function f( x)=2/x +x/3. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.
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Exemplo 9
Finding Local Maxima and Minima from a Graph
For the function f whose graph is shown in Figure 9, find all local maxima and minima. Figure 9
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Exemplo 10
Finding Absolute Maxima and Minima from a Graph
For the function f shown in Figure 14, find all absolute maxima and minima. Figure 14
Com figura
Algebra and Trigonometry
3.4 Composition of Functions
Exemplo 1
Performing Algebraic Operations on Functions
Find and simplify the functions ( g−f)( x) and ( g/f)( x), given f( x)=x−1 and g( x)=x^2 −1. Are they the same function?
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Exemplo 2
Determining whether Composition of Functions is Commutative
Using the functions provided, find f( g( x)) and g( f( x)). Determine whether the composition of the functions is commutative. f(x)=2x+1g(x)=3−x
Exemplo 3
Interpreting Composite Functions
The function c(s) gives the number of calories burned completing s sit-ups, and s(t) gives the number of sit-ups a person can complete in t minutes. Interpret c(s(3)).
Exemplo 4
Investigating the Order of Function Composition
Suppose f(x) gives miles that can be driven in x hours and g(y) gives the gallons of gas used in driving y miles. Which of these expressions is meaningful: f( g(y)) or g( f(x))?
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Exemplo 5
Using a Table to Evaluate a Composite Function
Using Table 1, evaluate f(g(3)) and g(f(3)). x f(x) g(x) 1 6 3 2 8 5 3 3 2 4 1 7 Table 1
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Exemplo 6
Using a Graph to Evaluate a Composite Function
Using Figure 1, evaluate f(g(1)). Figure 1
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Exemplo 7
Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input
Given f(t)=t^2 −t and h(x)=3x+2, evaluate f(h(1)).
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Exemplo 8
Finding the Domain of a Composite Function
Find the domain of ( f∘g)(x)wheref(x)=5/(x−1) andg(x)=4/(3x−2)
Exemplo 9
Finding the Domain of a Composite Function Involving Radicals
Find the domain of ( f∘g)(x) wheref(x)=√(x+2) and g(x)=√(3−x)
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Exemplo 10
Decomposing a Function
Write f(x)=√(5−x^2) as the composition of two functions.
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Algebra and Trigonometry
3.5 Transformation of Functions
Exemplo 1
Adding a Constant to a Function
To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 3 shows the area of open vents V (in square feet) throughout the day in hours after midnight, t.…
Com figura
Exemplo 2
Shifting a Tabular Function Vertically
A function f( x) is given in Table 2. Create a table for the function g(x)=f(x)−3. x 2 4 6 8 f(x) 1 3 7 11 Table 2
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Exemplo 3
Adding a Constant to an Input
Returning to our building airflow example from Figure 3, suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours…
Exemplo 4
Shifting a Tabular Function Horizontally
A function f(x) is given in Table 4. Create a table for the function g(x)=f(x−3). x 2 4 6 8 f(x) 1 3 7 11 Table 4
Exemplo 5
Identifying a Horizontal Shift of a Toolkit Function
Figure 8 represents a transformation of the toolkit function f(x)=x^2. Relate this new function g(x) to f(x), and then find a formula for g(x). Figure 8
Com figura
Exemplo 6
Interpreting Horizontal versus Vertical Shifts
The function G(m) gives the number of gallons of gas required to drive m miles. Interpret G(m)+10 and G(m+10).
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Exemplo 7
Graphing Combined Vertical and Horizontal Shifts
Given f(x)=| x |, sketch a graph of h(x)=f(x+1)−3.
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Exemplo 8
Identifying Combined Vertical and Horizontal Shifts
Write a formula for the graph shown in Figure 11, which is a transformation of the toolkit square root function. Figure 11
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Exemplo 9
Reflecting a Graph Horizontally and Vertically
Reflect the graph of s(t)=√t (a) vertically and (b) horizontally.
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Exemplo 10
Reflecting a Tabular Function Horizontally and Vertically
A function f(x) is given as Table 6. Create a table for the functions below. ⓐ g(x)=−f(x) ⓑ h(x)=f(−x) x 2 4 6 8 f(x) 1 3 7 11 Table 6
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Exemplo 11
Applying a Learning Model Equation
A common model for learning has an equation similar to k(t)=−2^(−t) +1, where k is the percentage of mastery that can be achieved after t practice sessions. This is a transformation of the function f(t)=2^t shown in…
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Exemplo 12
Determining whether a Function Is Even, Odd, or Neither
Is the function f(x)=x^3 +2x even, odd, or neither?
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Exemplo 13
Graphing a Vertical Stretch
A function P( t) models the population of fruit flies. The graph is shown in Figure 20. Figure 20 A scientist is comparing this population to another population, Q, whose growth follows the same pattern, but is twice…
Com figura
Exemplo 14
Finding a Vertical Compression of a Tabular Function
A function f is given as Table 10. Create a table for the function g(x)=1/2 f(x). x 2 4 6 8 f(x) 1 3 7 11 Table 10
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Exemplo 15
Recognizing a Vertical Stretch
The graph in Figure 22 is a transformation of the toolkit function f(x)=x^3. Relate this new function g(x) to f(x), and then find a formula for g(x). Figure 22
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Exemplo 16
Graphing a Horizontal Compression
Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, R, will progress in 1…
Exemplo 17
Finding a Horizontal Stretch for a Tabular Function
A function f(x) is given as Table 13. Create a table for the function g(x)=f( 1/2 x). x 2 4 6 8 f(x) 1 3 7 11 Table 13
Exemplo 18
Recognizing a Horizontal Compression on a Graph
Relate the function g(x) to f(x) in Figure 26. Figure 26
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Exemplo 19
Finding a Triple Transformation of a Tabular Function
Given Table 15 for the function f(x), create a table of values for the function g(x)=2f(3x)+1. x 6 12 18 24 f(x) 10 14 15 17 Table 15
Exemplo 20
Finding a Triple Transformation of a Graph
Use the graph of f( x) in Figure 27 to sketch a graph of k(x)=f( 1/2 x+1)−3. Figure 27
Com figura
Algebra and Trigonometry
3.6 Absolute Value Functions
Exemplo 1
Using Absolute Value to Determine Resistance
Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these…
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Exemplo 2
Writing an Equation for an Absolute Value Function Given a Graph
Write an equation for the function graphed in Figure 4. Figure 4
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Exemplo 3
Finding the Zeros of an Absolute Value Function
For the function f(x)=|4x+1|−7, find the values of x such that f(x)=0.
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Algebra and Trigonometry
3.7 Inverse Functions
Exemplo 1
Identifying an Inverse Function for a Given Input-Output Pair
If for a particular one-to-one function f(2)=4 and f(5)=12, what are the corresponding input and output values for the inverse function?
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Exemplo 2
Testing Inverse Relationships Algebraically
If f( x)=1/(x+2) and g( x)=1/x −2, is g=f^(−1)?
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Exemplo 3
Determining Inverse Relationships for Power Functions
If f(x)=x^3 (the cube function) and g(x)=1/3 x, is g=f^(−1)?
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Exemplo 4
Finding the Inverses of Toolkit Functions
Identify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on which each function is one-to-one, if any. The toolkit functions are reviewed in Table 2. We…
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Exemplo 5
Interpreting the Inverse of a Tabular Function
A function f(t) is given in Table 3, showing distance in miles that a car has traveled in t minutes. Find and interpret f^(−1) (70). t(minutes) 30 50 70 90 f( t)(miles) 20 40 60 70 Table 3
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Exemplo 6
Evaluating a Function and Its Inverse from a Graph at Specific Points
A function g(x) is given in Figure 5. Find g(3) and g^(−1) (3). Figure 5
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Exemplo 7
Inverting the Fahrenheit-to-Celsius Function
Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature. C=5/9 (F−32)
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Exemplo 8
Solving to Find an Inverse Function
Find the inverse of the function f( x)=2/(x−3) +4.
Exemplo 9
Solving to Find an Inverse with Radicals
Find the inverse of the function f(x)=2+√(x−4).
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Exemplo 10
Finding the Inverse of a Function Using Reflection about the Identity Line
Given the graph of f(x) in Figure 9, sketch a graph of f^(−1) (x). Figure 9
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Precalculus
1.1 Functions and Function Notation
Exemplo 1
Determining If Menu Price Lists Are Functions
The coffee shop menu, shown in Figure 2 consists of items and their prices. ⓐ Is price a function of the item? ⓑ Is the item a function of the price? Figure 2
Com figura
Exemplo 12
Reading Function Values from a Graph
Given the graph in Figure 7, ⓐ Evaluate f( 2). ⓑ Solve f( x)=4. Figure 7
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Exemplo 14
Applying the Vertical Line Test
Which of the graphs in Figure 12 represent(s) a function y=f( x)? Figure 12
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Exemplo 15
Applying the Horizontal Line Test
Consider the functions shown in Figure 12(a) and Figure 12(b). Are either of the functions one-to-one?
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Precalculus
1.3 Rates of Change and Behavior of Graphs
Exemplo 3
Computing Average Rate of Change from a Table
After picking up a friend who lives 10 miles away, Anna records her distance from home over time. The values are shown in Table 2. Find her average speed over the first 6 hours. t (hours) 0 1 2 3 4 5 6 7 D(t) (miles)…
Exemplo 6
Finding an Average Rate of Change as an Expression
Find the average rate of change of g( t)=t^2 +3t+1 on the interval [0,a]. The answer will be an expression involving a.
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Precalculus
1.6 Absolute Value Functions
Exemplo 1
Determine a Number within a Prescribed Distance
Describe all values x within or including a distance of 4 from the number 5.
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Exemplo 2
Resistance of a Resistor
Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these…
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Exemplo 3
Writing an Equation for an Absolute Value Function
Write an equation for the function graphed in Figure 5. Figure 5
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Exemplo 5
Solving an Absolute Value Equation
Solve 1=4| x−2 |+2.
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Exemplo 6
Solving an Absolute Value Inequality
Solve |x−5|<4.
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Exemplo 7
Using a Graphical Approach to Solve Absolute Value Inequalities
Given the function f(x)=−1/2 | 4x−5 |+3, determine the x- values for which the function values are negative.
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