Question

Feature
Three functions are graphed on the coordinate plane.The x-axis goes from negative 10 to 10 and the y-axis goes from negative 10 to 10. A function labeled f is the right half of a parabola with vertex left-parenthesis negative 5 comma 0 right-parenthesis. A function labeled g starts at point left-parenthesis 0 comma negative 5 right-parenthesis and curves up to the right. A function labeled h starts at point left-parenthesis 5 comma 0 right-parenthesis and curves down to the right.
Question
Use the graph to answer the question.
Which function, g or h, is the inverse function for function f?
(1 point)
Responses

the function g because the graphs of f and g are symmetrical about the x-axis
the function g because the graphs of f and g are symmetrical about the x -axis

the function g because the graphs of f and g are symmetrical about the line y = x
the function g because the graphs of f and g are symmetrical about the line y = x

the function h because the graphs of f and h are symmetrical about the line y = x
the function h because the graphs of f and h are symmetrical about the line y = x

the function h because the graphs of f and h are symmetrical about the x-axis
the function h because the graphs of f and h are symmetrical about the x -axis
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Answer 1

To determine which function, \( g \) or \( h \), is the inverse of function \( f \), we need to understand the concept of inverse functions in relation to their graphs.

Inverse functions have graphs that are symmetrical about the line \( y = x \). This means if you were to reflect the graph of one function over the line \( y = x \), you would get the graph of its inverse.

From your description:
- The function \( f \) is a right half of a parabola with its vertex at \( (-5, 0) \).
- The function \( g \) starts at \( (0, -5) \) and curves up to the right, which suggests it has the behavior of an increasing function.
- The function \( h \) starts at \( (5, 0) \) and curves down to the right, suggesting it has the behavior of a decreasing function.

Now let's assess the statements:
- The first two options mention symmetry about the x-axis, which is not relevant to inverse functions.
- The third option correctly states the necessary symmetry about the line \( y = x \).
- The fourth option is a repetition of the third option in wording.
- The fifth and sixth options claim symmetry about the line \( y = x \) for function \( h \), which is not correct based on your description of function \( h \).
- The last option refers to symmetry about the x-axis, which again is not relevant.

Based on this analysis, the correct response is:

**the function g because the graphs of f and g are symmetrical about the line y = x**.