Example 4 from Algebra and Trigonometry, 3.7 Inverse Functions

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 restrict the domain in such a fashion that the function assumes all y-values exactly once.

Constant	Identity	Quadratic	Cubic	Reciprocal
$f (x) = c$	$f (x) = x$	$f (x) = x^{2}$	$f (x) = x^{3}$	$f (x) = \frac{1}{x}$
Reciprocal squared	Cube root	Square root	Absolute value
$f (x) = \frac{1}{x^{2}}$	$f (x) = \sqrt[3]{x}$	$f (x) = \sqrt{x}$	$f (x) = \| x \|$

Table 2

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 constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no inverse.
The absolute value function can be restricted to the domain $[0, \infty),$ where it is equal to the identity function.
The reciprocal-squared function can be restricted to the domain $(0, \infty) .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

The domain of function $f$ is $(1, \infty)$ and the range of function $f$ is $(−∞, −2) .$ Find the domain and range of the inverse function.

How did it go?

Constant

Identity

Quadratic

Cubic

Reciprocal

f (x) = c

f (x) = x

f (x) = x^{2}

f (x) = x^{3}

f (x) = \frac{1}{x}

Reciprocal squared

Cube root

Square root

Absolute value

f (x) = \frac{1}{x^{2}}

f (x) = \sqrt[3]{x}

f (x) = \sqrt{x}

f (x) = | x |

Solution

Your turn

Solution

Your turn