Question

Inverse on the same set of axes. f(x)=x+9

Answer 1

The inverse function of
`\( f(x) = x + 9 \) is \( f^(-1)(x) = x - 9 \).`

Step-by-step explanation:

The inverse function undoes the operations of the original function. In this case, the original function \( f(x) = x + 9 \) adds 9 to the input. To find the inverse, we need to reverse this operation. Subtracting 9 from the output of the original function gives us the inverse function, \( f^{-1}(x) = x - 9 \).

To understand this conceptually, consider the process in two steps.
`If we start with a value \( x \), applying the original function \( f(x) = x + 9 \) would yield \( x + 9 \). Now, to undo this addition and retrieve the original \( x \), we subtract 9 from \( x + 9 \), resulting in \( x - 9 \). This is why the inverse function is \( f^(-1)(x) = x - 9 \).`

Graphically, the functions \( f(x) = x + 9 \) and its inverse
`\( f^(-1)(x) = x - 9 \)`are reflections of each other across the line \( y = x \), as is the case with any function and its inverse. The point (a, b) on \( f(x) \) corresponds to the point (b, a) on
`\( f^(-1)(x) \)`, reinforcing the reversal of roles between the input and output.