Exponential and Logarithmic Functions: Part One — Test

Question 19 • Advanced Algebra: Concepts and Connections

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

Answer

Image with description On a coordinate plane with x and y axes marked from negative 5 to 5, 2 lines are shown. A curve labeled f of (x) has a positive slope and goes through (negative 4 comma negative 3), (0 comma negative 1), and (2 comma 0). Another curve labeled inverse of f of (x) has a positive slope and goes through (negative 2 comma negative 2), (negative 1comma 0), and (0 comma 2).

Image with description On a coordinate plane with x and y axes marked from negative 5 to 5, 2 v-shaped graphs are shown. A v-shaped graph labeled f of (x) opens up and goes through (negative 1 comma 5), has a vertex at (1 comma 1), and goes through (3 comma 5). Another v-shaped graph labeled inverse of f of (x) opens right and goes through (5 comma 3), has a vertex at (1 comma 1), and goes through (3 comma 0).

Image with description On a coordinate plane with x and y axes marked from negative 5 to 5, two curved graphs are shown. A curve labeled f of (x) sharply increases from (negative 0 point 5 comma negative 5) to (0 comma negative 2) and then sharply changes directions and curves down to (0 point 5 comma negative 3 point 5). The curve then sharply changes direction and goes up through (1 comma 0). Another curve labeled inverse of f of (x) curves slightly up from (negative 5 comma negative 0 point 5) to (negative 2 comma 0). The curve then sharply changes directions and go to the left toward (negative 3 point 5 comma 1). The curve then changes directions and goes through (0 comma 1) and gradually increases.

Image with description On a coordinate plane with x and y axes marked from negative 5 to 5, 2 parabolas are shown. A curve labeled f of (x) opens down and goes through (negative 1 comma negative 5), has a vertex at (0 comma 1), and goes through (1 comma negative 5). Another curve labeled inverse of f of (x) opens to the left and goes through (negative 5 comma 1), has a vertex at (1 comma 0), and goes through (negative 5 comma negative 1).

Explanation

A function has an inverse that is also a function only if it is one-to-one, meaning no horizontal line crosses its graph more than once. In choice A, the graph of f is an increasing line, so every horizontal line crosses it at most once; its inverse is also a function. The other graphs include turning points, so some horizontal lines cross them more than once.

AI-written and checked against the verified answer.

More from this Course

Test

Test

Test