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Exercises Functions

Example 4 from Algebra and Trigonometry, 3.5 Transformation of Functions

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

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The formula $g (x) = f (x - 3)$ tells us that the output values of $g$ are the same as the output value of $f$ when the input value is 3 less than the original value. For example, we know that $f (2) = 1.$ To get the same output from the function $g,$ we will need an input value that is 3 larger. We input a value that is 3 larger for $g (x)$ because the function takes 3 away before evaluating the function $f .$

$\begin{matrix} g (5) & = & f (5 - 3) \\ = & f (2) \\ = & 1 \end{matrix}$
We continue with the other values to create Table 5.

$x$ 5 7 9 11

$x - 3$ 2 4 6 8

$f (x - 3)$ 1 3 7 11

$g (x)$ 1 3 7 11

Table 5

The result is that the function $g (x)$ has been shifted to the right by 3. Notice the output values for $g (x)$ remain the same as the output values for $f (x),$ but the corresponding input values, $x,$ have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.

How did it go?

PreviousAdding a Constant to an Input Next Identifying a Horizontal Shift of a Toolkit Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 3.5, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.