Warm Up Solutions

Determine the inverse function.

$m (x) = - \frac{1}{2} (x - 16) + 10$

$\begin{array}{rcl} y & = & - \frac{1}{2} (x - 16) + 10 \\ x & = & - \frac{1}{2} (y - 16) + 10 \\ x - 10 & = & - \frac{1}{2} y + 8 \\ x - 18 & = & - \frac{1}{2} y \\ y & = & - 2 (x - 18) \\ or y & = & - 2 x + 36 \end{array}$

Note

Q: What is the domain and range for the function and its inverse?
A: The domain and range for both are all real numbers.

$g (x) = 4 {(x + 9)}^{2} - 11$

$\begin{array}{rcl} y & = & 4 {(x + 9)}^{2} - 11 \\ x & = & 4 {(y + 9)}^{2} - 11 \\ \frac{x + 11}{4} & = & \frac{4 {(y + 9)}^{2}}{4} \\ \pm \sqrt{\frac{x + 11}{4}} & = & \sqrt{{(y + 9)}^{2}} \\ \pm \frac{1}{2} \sqrt{x + 11} & = & y + 9 \\ y & = & \pm \frac{1}{2} \sqrt{x + 11} - 9 \end{array}$

Note

Q: What restrictions are needed in the inverse of g(x)? Explain.
A: The values of x must be greater than or equal to $- 11$ so that the square root is a real number.

Q: What is the domain and range for the function and its inverse?
A: For g(x) the domain is all real numbers and the range is greater than or equal to –11. For the inverse, the domain is greater than or equal to –11, and the range is all real numbers.

Remember to include the +/– symbol in your answer to represent the complete inverse of the function.