Exponential and Logarithmic Functions: Part One — Test

Question 12 • Advanced Algebra: Concepts and Connections

Which graph shows a function whose inverse is also a function?

Answer

Image with description A Cartesian coordinate plane with horizontal x-axis and vertical y-axis, both grid-lined and ranges from negative 5 to 5 with an increment of 1. The function f(x) is represented by a red curve starting at the origin "0 comma 0", curving steeply upward to "1 comma 1", and then flattening out as it passes through "4 comma 2" before continuing to the right with an arrowhead. The inverse function f superscript (-1) (x) is represented by a blue curve starting at the origin "0 comma 0", staying relatively flat until it intersects the red curve at "1 comma 1", and then curving steeply upward through "2 comma 4" before continuing upward with an arrowhead.

Image with description A Cartesian coordinate plane with horizontal x-axis and vertical y-axis, both grid-lined and ranges from negative 5 to 5 with an increment of 1. The function f(x) is a blue, sharp-peaked curve with its highest point (vertex) at "1 comma 3". From the vertex, one ray extends downward and to the left, passing through "0 comma negative 3" with an arrowhead pointing down. The other ray extends downward and to the right, passing through "2 comma negative 3" with an arrowhead pointing down.The inverse function f superscript {-1} (x) is a red, sideways sharp-peaked curve with its rightmost point (vertex) at "3 comma 1". From this vertex, one ray extends to the left and slightly upward, passing through "negative 3 comma 2" with an arrowhead pointing left. The other ray extends to the left and slightly downward, passing through "negative 3 comma 0" with an arrowhead pointing left.

Image with description A Cartesian coordinate plane with horizontal x-axis and vertical y-axis, both grid-lined and ranges from negative 5 to 5 with an increment of 1. The function f(x) is a blue parabola that opens upward with its vertex located on the y-axis at "0 comma negative 2". It passes through the x-axis near "negative 1 comma 0" and "1 point 0", and extends upward through "negative 1 point 2 comma 1" and "1 point 2 comma 1" with arrowheads pointing up on both ends.The inverse function f superscript {-1}(x) is a red parabola that opens to the right with its vertex located on the x-axis at " negative 2 comma 0". It passes through the y-axis near "0 comma negative 1" and "0 comma 1", and extends horizontally to the right through "1 comma negative 1 point 2" and "1 comma 1 point 2" with arrowheads pointing right on both ends.

Image with description A Cartesian coordinate plane with horizontal x-axis and vertical y-axis, both grid-lined and ranges from negative 5 to 5 with an increment of 1. The function f(x) is a blue curve that enters from the bottom-left, passing near "negative 1 comma negative 3" and rising steeply. It creates a local maximum at "0 comma 2", dips down to a local minimum at "1 comma 1", and then rises sharply toward the top right, passing through "1 point 2 comma 4" with an arrowhead pointing upward. The inverse function f superscript {-1} (x) is a red curve that enters from the far left near " negative 4 comma negative 1" and proceeds horizontally to the right. It curves upward and backward to form a local maximum on the right side at "2 comma 0", snakes back to the left to hit a local minimum at "0 point 6 comma 1", and then levels off to the right, exiting past "4 comma 1 point 2" with an arrowhead pointing right.

Explanation

A function has an inverse that is also a function when it is one-to-one: each output comes from only one input. In graph A, the red curve keeps increasing, so every horizontal line meets it at most once; its inverse is therefore a function. The graphs in B and C repeat output values, and D has turning points, so their inverses would not be functions.

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