+25 Function g is defined as follows:
\[g(x) = \left\{
\begin{array}{cl}
5x^2 & \text{if } x \le -3, \\
21+x& \text{if }-3 < x \le 10, \\
2-\sqrt{x}& \text{if }x > 10.
\end{array}\right.\]
This function has an inverse
What is g^-1(7)?
The inverse of a function is another function that reverses the output of the original function. In this case, the inverse of g(x) would be a function that takes in the output of g(x) and returns the input.
To find the inverse of g(x), we can start by writing it as a set of three equations:
g(x) = 5x^2 if x <= -3 g(x) = 21 + x if -3 < x <= 10 g(x) = 2 - sqrt(x) if x > 10
We can then solve each equation for x in terms of g(x). This gives us the following three equations:
x = sqrt(g(x)/5) if x <= -3 x = g(x) - 21 if -3 < x <= 10 x = 1/sqrt(2 - g(x)) if x > 10
The inverse of g(x) is the function that takes in the output of g(x) and returns the corresponding input. In this case, the inverse of g(x) is a piecewise function that is made up of the three equations above.
To find g inverse of 7, we can simply plug 7 into each of the three equations and see which one gives us a valid input. In this case, the only equation that gives us a valid input is the second equation, so g inverse of 7 is 7 - 21 = -14.
