Question

and
are inverse functions, and their graphs are drawn on the same coordinate plane with the same scale on both the x- and y-axes. Which one of the following is true?
Responses


is the same as
translated up 3 units.
g of x is the same as f of x translated up 3 units.

By rotating
clockwise around the origin, you would get
.
By rotating
clockwise around the origin, you would get g of x.

If you reflect
around the line
, it will land on
.

Answer 1

When \( f(x) \) and \( g(x) \) are inverse functions, certain properties hold true based on their definitions and the nature of inverse functions. Let's analyze the given statements:

1. **\( g(x) \) is the same as \( f(x) \) translated up 3 units.**
- This statement is false, as translating \( f(x) \) up by 3 units would not yield \( g(x) \). Instead, the correct relationship is that \( g(x) \) will reflect \( f(x) \) over the line \( y = x \).

2. **\( g(x) \) is the same as \( f(x) \) translated up 3 units.**
- This statement is also false for the same reason as above.

3. **By rotating \( g(x) \) clockwise around the origin, you would get \( f(x) \).**
- This statement is false. Rotating a function clockwise around the origin changes the relationship between the inputs and outputs in a way that does not produce the inverse function.

4. **By reflecting \( f(x) \) around the line \( y = x \), it will land on \( g(x) \).**
- This statement is true. Reflecting a function \( f(x) \) across the line \( y = x \) gives you its inverse \( g(x) \).

From this analysis, the true statement is:

**If you reflect \( f \) around the line \( y = x \), it will land on \( g \).**