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

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

GPT-4o mini · Answer

To determine which function is the inverse of function \( f \), we need to remember that inverse functions are symmetrical about the line \( y = x \).

Given the descriptions:

Function \( f \) is the right half of a parabola with vertex at \((-5, 0)\).
Function \( g \) starts at \((0, -5)\) and curves upward to the right.
Function \( h \) starts at \((5, 0)\) and curves downward to the right.

If function \( g \) and function \( f \) are inverses, then each point \((a, b)\) on the graph of \( f \) should correspond to a point \((b, a)\) on the graph of \( g\) (and vice versa).

Based on these transformations:

\( f \) has points that would lie in the region where \( x \) values are negative or zero and \( y \) values are positive (since it is the right half of a parabola).
Therefore, for \( g \) to be the inverse of \( f \), it should map those \( y \) values back to the corresponding \( x \) values.
Function \( h\) has a starting point at \( (5, 0) \), which does not correspond to the range of \( f\).

From the descriptions, \( g \) seems to start mapping values similar to what would be required to correspond with \( f \).

Thus the correct answer would be:

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