Question

if a function is translated 3 units to the right, in which direction will the inverse of the function be transformed? (1 point)

• the inverse of the given function will be translated 3 units down.
• the inverse of the given function will be translated 3 units to the right.
• the inverse of the given function will be translated 3 units to the left
• the inverse of the given function will be translated 3 units up

Explanation:

To determine the transformation of the inverse function when the original function is translated 3 units to the right, we analyze the relationship between a function and its inverse.

1. Recall that the graph of an inverse function \( y = f^{-1}(x) \) is the reflection of the graph of \( y = f(x) \) over the line \( y = x \).
2. A horizontal translation of the original function (e.g., \( f(x) \to f(x - 3) \) for a rightward shift) affects the inverse function's graph by a corresponding vertical or horizontal shift, but due to the reflection over \( y = x \), a horizontal shift in the original function corresponds to a vertical shift in the inverse, and vice versa. Wait, actually, let's think in terms of points. If the original function \( y = f(x) \) has a point \( (a, b) \), the inverse function \( y = f^{-1}(x) \) has the point \( (b, a) \).
3. If the original function is shifted 3 units to the right, the new function is \( y = f(x - 3) \). So a point \( (a + 3, b) \) is on the new function. The inverse of this new function will have the point \( (b, a + 3) \). Comparing to the original inverse function's point \( (b, a) \), this is a shift of 3 units up? Wait, no, maybe I mixed up. Wait, no—wait, let's take a concrete example. Let \( f(x) = x \), so its inverse is \( f^{-1}(x) = x \). If we shift \( f(x) \) 3 units to the right, we get \( f(x - 3) = x - 3 \). The inverse of \( x - 3 \) is \( y = x + 3 \), which is a shift of 3 units up? Wait, no, that contradicts. Wait, no—wait, \( f(x) = x \), shift right 3: \( f(x - 3) = x - 3 \). To find the inverse, solve \( y = x - 3 \) for \( x \): \( x = y + 3 \), so the inverse is \( f^{-1}(y) = y + 3 \), or \( f^{-1}(x) = x + 3 \). So the original inverse was \( f^{-1}(x) = x \), and the new inverse is \( f^{-1}(x) = x + 3 \), which is a shift of 3 units up. Wait, but the options given are down, right, left, up. Wait, maybe my example is wrong. Wait, no—wait, maybe I confused horizontal and vertical shifts. Wait, another example: let \( f(x) = 2x \), inverse is \( f^{-1}(x) = x/2 \). Shift \( f(x) \) 3 units right: \( f(x - 3) = 2(x - 3) = 2x - 6 \). The inverse of this is found by solving \( y = 2x - 6 \) for \( x \): \( x = (y + 6)/2 = y/2 + 3 \), so the inverse is \( f^{-1}(x) = x/2 + 3 \). The original inverse was \( f^{-1}(x) = x/2 \), so the new inverse is \( x/2 + 3 \), which is a shift of 3 units up. Wait, but the options include "3 units up" as an option (the last one). But wait, maybe I made a mistake earlier. Wait, the question is: If a function is translated 3 units to the right, in which direction will the inverse of the function be transformed? Wait, maybe the key is that when you reflect over \( y = x \), a horizontal shift in the original function (right) corresponds to a vertical shift in the inverse (up), and a vertical shift in the original corresponds to a horizontal shift in the inverse. Wait, but let's re-examine the options. Wait, the options are: down, right, left, up. Wait, in my first example, shifting the original function right 3 led to the inverse shifting up 3. But let's check the options. The last option is "The inverse of the given function will be translated 3 units up". But wait, maybe I messed up. Wait, no—wait, let's think again. Suppose the original function is \( y = f(x) \), inverse is \( x = f(y) \), or \( y = f^{-1}(x) \). If we shift the original function 3 units right, we have \( y = f(x - 3) \). To find the inverse, solve for \( x \): \( x - 3 = f^{-1}(y) \), so \( x = f^{-1}(y) + 3 \), so the inverse function is \( y = f^{-1}(x) + 3 \), which is a shift of 3 units up. Therefore, the inverse function is translated 3 units up. Wait, but let's check the options. The options are:

1. The inverse of the given function will be translated 3 units down.

2. The inverse of the given function will be translated 3 units to the right.

3. The inverse of the given function will be translated 3 units to the left.

4. The inverse of the given function will be translated 3 units up.

So the correct answer should be the fourth option: "The inverse of the given function will be translated 3 units up". Wait, but let me confirm with another example. Let \( f(x) = 2x + 1 \), inverse is \( f^{-1}(x) = (x - 1)/2 \). Shift \( f(x) \) 3 units right: \( f(x - 3) = 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5 \). Inverse of \( 2x - 5 \): solve \( y = 2x - 5 \) for \( x \): \( 2x = y + 5 \), \( x = (y + 5)/2 = y/2 + 5/2 \). So the inverse is \( f^{-1}(x) = x/2 + 5/2 \). The original inverse was \( (x - 1)/2 = x/2 - 1/2 \). The difference is \( (x/2 + 5/2) - (x/2 - 1/2) = 3 \), so it's a shift of 3 units up. Yes, that confirms it. So the inverse function is translated 3 units up.

Answer:

The inverse of the given function will be translated 3 units up (the fourth option, assuming the options are ordered as: 1. down, 2. right, 3. left, 4. up). So the correct option is:

The inverse of the given function will be translated 3 units up (the last option in the list).