Lesson Objectives

- Learn how to determine if a function is one-to-one using an algebraic method

## How to Determine if a Function is One-to-One Using an Algebraic Method

In the last lesson, we learned how to determine if a function was a one-to-one function using the horizontal line test. Now, we will show how to accomplish the same task using an algebraic method.

First and foremost, we can say that a function is one-to-one if and only if: $$f(a)=f(b)$$ implies that: $$a=b$$ We can use this to develop a simple test. Let's work through a few examples.

Example #1: Determine if the following function is one-to-one. $$f(x)=3x^2 - 5$$ First, we will find f(a) and f(b). $$f(a)=3a^2 - 5$$ $$f(b)=3b^2 - 5$$ If f(a) = f(b), then a = b.

Let's set these two expression equal to each other and see if a = b: $$3a^2 - 5=3b^2 - 5$$ Add 5 to both sides: $$3a^2 - 5 + 5=3b^2 - 5 + 5$$ $$3a^2=3b^2$$ Divide both sides by 3: $$\frac{3a^2}{3}=\frac{3b^2}{3}$$ $$a^2=b^2$$ Does a equal b? Not always, so we would say this function isn't one-to-one. We can show this more clearly if we solve for a: $$a=\pm b$$ Let's try another example.

Example #2: Determine if the function is one-to-one. $$f(x)=\frac{5}{x - 9}$$ First, we will find f(a) and f(b). $$f(a)=\frac{5}{a - 9}$$ $$f(b)=\frac{5}{b - 9}$$ Let's set these two expressions equal to each other: $$\frac{5}{a - 9}=\frac{5}{b - 9}$$ Cross Multiply: $$5(b-9)=5(a - 9)$$ Divide both sides by 5: $$b - 9=a - 9$$ Add 9 to both sides: $$b=a$$ Since b = a or a = b, we can say this function is one-to-one.

First and foremost, we can say that a function is one-to-one if and only if: $$f(a)=f(b)$$ implies that: $$a=b$$ We can use this to develop a simple test. Let's work through a few examples.

Example #1: Determine if the following function is one-to-one. $$f(x)=3x^2 - 5$$ First, we will find f(a) and f(b). $$f(a)=3a^2 - 5$$ $$f(b)=3b^2 - 5$$ If f(a) = f(b), then a = b.

Let's set these two expression equal to each other and see if a = b: $$3a^2 - 5=3b^2 - 5$$ Add 5 to both sides: $$3a^2 - 5 + 5=3b^2 - 5 + 5$$ $$3a^2=3b^2$$ Divide both sides by 3: $$\frac{3a^2}{3}=\frac{3b^2}{3}$$ $$a^2=b^2$$ Does a equal b? Not always, so we would say this function isn't one-to-one. We can show this more clearly if we solve for a: $$a=\pm b$$ Let's try another example.

Example #2: Determine if the function is one-to-one. $$f(x)=\frac{5}{x - 9}$$ First, we will find f(a) and f(b). $$f(a)=\frac{5}{a - 9}$$ $$f(b)=\frac{5}{b - 9}$$ Let's set these two expressions equal to each other: $$\frac{5}{a - 9}=\frac{5}{b - 9}$$ Cross Multiply: $$5(b-9)=5(a - 9)$$ Divide both sides by 5: $$b - 9=a - 9$$ Add 9 to both sides: $$b=a$$ Since b = a or a = b, we can say this function is one-to-one.

#### Skills Check:

Example #1

Determine if the function is one-to-one. $$f(x)=13x^4 - 3$$

Please choose the best answer.

A

Yes

B

No

Example #2

Determine if the function is one-to-one. $$f(x)=\frac{3}{x}- 11$$

Please choose the best answer.

A

Yes

B

Yes

Example #3

Determine if the function is one-to-one. $$f(x)=|2x - 7| + 9$$

Please choose the best answer.

A

Yes

B

No

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