Explore: Verifying Inverses Solutions

Verifying Inverses Solutions

Two functions, f and g, are inverses of one another when $f (a) = b$ and $g (b) = a .$
Any ordered pair $(a, b)$ in the given function is $(b, a) in the inverse$ .

Example 3

Verify that the functions f (x) and g(x) are inverses of one another using $f (- 12)$ .

$f (x) = 8 {(x + 14)}^{3} - 18$

$g (x) = \frac{\sqrt[3]{x + 18}}{2} - 14$

Plan

Find $f (a) = b$
Find g(b)

Implement

$x = - 12 f (- 12) = 8 ({(- 12) + 14)}^{3} - 18 f (- 12) = 8 {(2)}^{3} - 18$

$f (- 12) = 8 (8) - 18 f (- 12) = 64 - 18 f (- 12) = 46$

Explain

Substitute
Evaluate

Note

Using –12 means you will cube 2, which is more manageable than a very large number.

It is recommended that you write down substitution and do the calculations with a calculator. Assessing inverses is the objective, not order of operations.

$g (46) = \frac{\sqrt[3]{46 + 18}}{2} - 14 g (46) = \frac{\sqrt[3]{64}}{2} - 14 g (46) = \frac{4}{2} - 14 = 2 - 14 g (46) = - 12$

The functions are inverses of one another by definition since $f (- 12) = 46$ and $g (46) = - 12 .$

Example 4

Verify that the given functions are inverses of one another for h(1). Explain your reasoning.

$h (x) = 4 x - 8$

$h (1) = 4 (1) - 8 h (1) = 4 - 8 h (1) = - 4$

$j (x) = \frac{1}{4} x + 8$

$j (- 4) = \frac{1}{4} (- 4) + 8 j (- 4) = - 1 + 8 j (- 4) = 7$

The functions h(x) and j(x) are not inverses of one another because $h (1) = - 4$ and $j (- 4) = 7$ . By definition of inverses, j (– 4) would need to be 1.

Note

Remember that the $f (a) = b$ , so b is substituted into the second equation.

For this example, the instructions state which value to use (1). If the directions do not provide a value, you can use a value of your choice.

Example 5

Verify which function, j(x) or k(x), is the inverse of g(x).

$g (x) = \sqrt{x - 3} - 1 g (3) = - 1$

$j (x) = {(x + 3)}^{2} + 1, \{x | x \geq - 3\}$

$j (- 1) = ({(- 1) + 3)}^{2} + 1 j (- 1) = {(2)}^{2} + 1 j (- 1) = 5$

$k (x) = {(x + 1)}^{2} + 3, \{x | x \geq - 1\}$

$k (- 1) = ({(- 1) + 1)}^{2} + 3 k (- 1) = 0 + 3 k (- 1) = 3$

The function $k (x)$ is the inverse of $g (x)$ because $g (3) = - 1$ and $k (- 1) = 3 .$

Note

You will learn more about restrictions on original functions when there is an inequality beside them (in this example $k (x)$ ) in the next part of the lesson.