# Inverse Functions: One to One

Not all functions have inverse functions. The graph of inverse functions are reflections over the line y = x. This means that each x-value must be matched to one and only one y-value. Functions that meet this criteria are called one-to one functions.

DEFINITION OF ONE-TO-ONE:

A function is said to be one-to-one if each x-value corresponds to exactly one y-value.

A function f has an inverse function, f -1, if and only if f is one-to-one.

A quick test for a one-to-one function is the horizontal line test. If a horizontal line intersects the graph of the function in more than one place, the functions is NOT one-to-one.

HORIZONTAL LINE TEST:

A function f is one-to-one and has an inverse function if and only if no horizontal line intersects the graph of f at more than one point.

Let's use this characteristic to determine if a function has an inverse.

Example 1: Use the Horizontal Line Test to determine if f(x) = 2x3 - 1 has an inverse function.
 Step 1: Sketch the graph of the function. Step 2: Apply the Horizontal Line Test. Visualize multiple horizontal lines and look for places where the graph is intersected more than once. No horizontal line intersects the graph in more than one place and thus the function has an inverse. Example 2: Sketch the graph represented by the points and determine if it has an inverse function.

 x -3 -2 -1 0 1 2 3 y 5 0 -3 -4 -3 0 5

Step 1: Sketch the graph of the function. Step 2: Apply the Horizontal Line Test.

Visualize multiple horizontal lines and look for places where the graph is intersected more than once.

Several horizontal lines intersect the graph in two places. Thus the function is not a one-to-one and does not have an inverse.

Also notice that if the ordered pairs are switched, this results in repeating x-values and a function cannot have repeating x-values.

 x 5 0 -3 -4 -3 0 5 y -3 -2 -1 0 1 2 3 Related Links: Math algebra Inverse Functions: Finding Inverse Functions Analytically Conics: Classifying from General Equation Pre Calculus

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