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 q is the left half of a parabola opening upwards with vertex left-parenthesis 0 comma negative 5 right-parenthesis. A function labeled r starts at point left-parenthesis negative 5 comma 0 right-parenthesis and curves down to the right. A function labeled s is the left half of a downward facing parabola with vertex left-parenthesis 0 comma 1.5 right-parenthesis.

Question

Question
Use the image to answer the question.
Which function, r or s, is the inverse function for function q?
(1 point)
Responses

the function r because the graphs of q and r are symmetrical about the y-axis
the function r because the graphs of q and r are symmetrical about the y -axis

the function r because the graphs of q and r are symmetrical about the line y = x
the function r because the graphs of q and r are symmetrical about the line y = x

the function s because the graphs of q and s are symmetrical about the line y = x
the function s because the graphs of q and s are symmetrical about the line y = x

neither function

Answer 1

To determine which function, \( r \) or \( s \), is the inverse of function \( q \), we need to consider the properties of inverse functions. Two functions \( f(x) \) and \( g(x) \) are inverses of each other if the graphs of \( f \) and \( g \) are symmetric about the line \( y = x \).

Given the descriptions:

• Function \( q \) is the left half of a parabola opening upwards with a vertex at (0, -5).
• Function \( r \) starts at point (-5, 0) and curves down to the right.
• Function \( s \) is the left half of a downward-facing parabola with a vertex at (0, 1.5).

Since \( q \) is an upward-opening parabola, its behavior would likely suggest that its inverse could be a function that mirrors it about the line \( y = x \).

Function \( r \), which curves down to the right starting from (-5, 0), does not follow the expected behavior of being the inverse of \( q \) since its overall shape does not reflect the symmetrical property.

Function \( s \) is a downward-facing parabola and is the left half of that parabola. Since \( q \) is the left half of an upward parabola, \( s \) could be thought of as the corresponding inverse shape.

Thus, the most fitting answer is:

the function s because the graphs of q and s are symmetrical about the line y = x.