It's time to practice! For each function, find its inverse.
Practice 1
Practice 2
Practice 3
Find the inverse of f(x) = 7x2 − 16.
| f(x) = 7x2 − 16 |
| y = 7x2 − 16 |
| x = 7y2 − 16 |
| 7y2 = x + 16 |
| y2 = \(\small\mathsf{ \frac{x+16}{7} }\) |
| \(\small\mathsf{ y^{2} }\) = \(\small\mathsf{ \pm \sqrt{\frac{x+16}{7}} }\) |
| y = \(\small\mathsf{ \pm \sqrt{\frac{x+16}{7}} }\) |
Since this is an even root, we must find out what value of x will make \(\small\mathsf{ \frac{x+16}{7} }\) greater than zero so the answer is a real number.
| \(\small\mathsf{ \frac{x+16}{7} }\) > 0 (Multiply both sides by 7) |
| x + 16 > 0 |
| x > -16 |
| f-1(x) = \(\small\mathsf{ \pm \sqrt{\frac{x+16}{7}} }\) when x > -16 |
Find the inverse of f(x) = x3 + 23.
| f(x) = x3 + 23 |
| y = x3 + 23 |
| x = y3 + 23 |
| y3 = x − 23 |
| \(\small\mathsf{ \sqrt[3]{y^{3}} }\) = \(\small\mathsf{ \sqrt[3]{x-23} }\) |
| y = \(\small\mathsf{ \sqrt[3]{x-23} }\) |
| f-1(x) = \(\small\mathsf{ \sqrt[3]{x-23} }\) |
Find the inverse of f(x) = -x4 + 18.
| f(x) = -x4 + 18 |
| y = -x4 + 18 |
| x = -y4 + 18 |
| y4 = -x + 18 |
| \(\small\mathsf{ \sqrt[4]{y^{4}} }\) = \(\small\mathsf{ \pm \sqrt[4]{-x+18} }\) |
| y = \(\small\mathsf{ \pm \sqrt[4]{-x+18} }\) |
The fourth root is an even number so (-x + 8) must be greater than zero.
-x + 18 > 0
-x > -18
(Divide both sides by -1. This flips the > to <.)
x < 18
f-1(x) = \(\small\mathsf{ \pm \sqrt[4]{-x+18} }\) when x < 18
