Question

1. let $f(x) = 4^x$. then, choose the correct answers for the question below.

when $x = -1$, then the coordinates for the function = $(-4, -1)$ and the coordinates of the inverse = $(1, -4)$.

1. let $f(x) = 4^x$. then, choose the correct answers for the question below.

when $x = 2$, then the coordinates for the function = and the coordinates of the inverse =

Explanation:

For the function \( f(x) = 4^x \) when \( x = 2 \):

Step 1: Find the \( y \)-value for the function

To find the \( y \)-value (i.e., \( f(x) \)) when \( x = 2 \), substitute \( x = 2 \) into the function \( f(x) = 4^x \).

$$ f(2) = 4^2 = 16 $$

So, the coordinates for the function when \( x = 2 \) are \( (2, 16) \).

Step 2: Find the coordinates of the inverse function

The inverse of a function \( f(x) \) swaps the \( x \)- and \( y \)-coordinates of the original function. So, if the original function has a point \( (a, b) \), the inverse function will have the point \( (b, a) \).
For the point \( (2, 16) \) on the original function, the corresponding point on the inverse function will be \( (16, 2) \).

For the inverse function part (assuming we need to find the coordinates of the inverse function when the original function has \( (2, 16) \)):

Step 1: Recall the property of inverse functions

The inverse function \( f^{-1}(x) \) of a function \( f(x) \) satisfies the property that if \( f(a) = b \), then \( f^{-1}(b) = a \). Geometrically, this means that the point \( (a, b) \) on \( f(x) \) corresponds to the point \( (b, a) \) on \( f^{-1}(x) \).

Step 2: Apply the property to the given \( x = 2 \)

We found that for \( f(x) = 4^x \), when \( x = 2 \), \( f(2) = 16 \), so the point on \( f(x) \) is \( (2, 16) \). By the property of inverse functions, the point on \( f^{-1}(x) \) will be \( (16, 2) \).

Answer:

s:

• Coordinates for the function when \( x = 2 \): \( (2, 16) \)
• Coordinates of the inverse function: \( (16, 2) \)