Lesson Objectives

- Demonstrate an understanding of the concept of a one-to-one function
- 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. Recall that a one-to-one function has the following properties:

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 expressions 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: $$\require{cancel}\frac{\cancel{5}(b-9)}{\cancel{5}}=\frac{\cancel{5}(a - 9)}{\cancel{5}}$$ $$b - 9=a - 9$$ Add 9 to both sides: $$b - 9 + 9=a - 9 + 9$$ $$b=a$$ Since b = a or a = b, we can say this function is one-to-one.

Example #3: Determine if the function is one-to-one. $$f(x)=|x + 5|$$ First, we will find f(a) and f(b). $$f(a)=|a + 5|$$ $$f(b)=|b + 5|$$ Let's set these two expressions equal to each other: $$|a + 5|=|b + 5|$$ This turns into two cases. Recall that two expressions have the same absolute value if they are equal or if they are negatives.

Case 1 (Drop the absolute value bars): $$a + 5=b + 5$$ Subtract 5 away from each side: $$a + 5 - 5=b + 5 - 5$$ $$a=b$$ Case 2 (Drop the absolute value bars, change one expression into its opposite): $$-(a + 5)=b + 5$$ $$-a - 5=b + 5$$ Add 5 to both sides: $$-a - 5 + 5=b + 5 + 5$$ $$-a=b + 10$$ Multiply both sides by -1: $$-a \cdot -1=b \cdot -1 + 10 \cdot -1$$ $$a=-b - 10$$ Our result shows that our function is not a one-to-one function. We get two separate answers joined by the keyword "or": $$a=b$$ $$\text{or}$$ $$a=-b - 10$$

- Each element in the domain corresponds to a unique element in the range
- For each x-value, there is only one corresponding y-value

- No two different elements in the domain correspond to the same element in the range
- For each y-value, there is only one corresponding x-value

### Testing for a One-to-One Function

- Graphical Approach (Horizontal Line Test)
- If no horizontal line intersects the graph of a function at more than one point, the function is one-to-one

- Algebraic Method
- Assume x
_{1}and x_{2}are any two elements in the domain - Set f(x
_{1}) equal to f(x_{2}) and solve for x_{1}and x_{2} - If x
_{1}and x_{2}turn out to always be the same (equal), the function is one-to-one. Otherwise, it is not a one-to-one function

- Assume x
- Derivative Method (Used with Calculus)
- If the derivative of the function is always positive or always negative in the domain, the function is one-to-one
- If the derivative changes sign, the function is not one-to-one
- We can also say the function is one-to-one if it either increases or decreases on its entire domain

### Algebraic Method for a One-to-One Function

First and foremost, we can say that a function is one-to-one if and only if: $$f(a)=f(b)$$ implies that: $$a=b$$ In other words, if the y-values or function values are the same (f(a) = f(b)), the function can only be one-to-one if the x-values are the same a = b (a and b represent the same value).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 expressions 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: $$\require{cancel}\frac{\cancel{5}(b-9)}{\cancel{5}}=\frac{\cancel{5}(a - 9)}{\cancel{5}}$$ $$b - 9=a - 9$$ Add 9 to both sides: $$b - 9 + 9=a - 9 + 9$$ $$b=a$$ Since b = a or a = b, we can say this function is one-to-one.

Example #3: Determine if the function is one-to-one. $$f(x)=|x + 5|$$ First, we will find f(a) and f(b). $$f(a)=|a + 5|$$ $$f(b)=|b + 5|$$ Let's set these two expressions equal to each other: $$|a + 5|=|b + 5|$$ This turns into two cases. Recall that two expressions have the same absolute value if they are equal or if they are negatives.

Case 1 (Drop the absolute value bars): $$a + 5=b + 5$$ Subtract 5 away from each side: $$a + 5 - 5=b + 5 - 5$$ $$a=b$$ Case 2 (Drop the absolute value bars, change one expression into its opposite): $$-(a + 5)=b + 5$$ $$-a - 5=b + 5$$ Add 5 to both sides: $$-a - 5 + 5=b + 5 + 5$$ $$-a=b + 10$$ Multiply both sides by -1: $$-a \cdot -1=b \cdot -1 + 10 \cdot -1$$ $$a=-b - 10$$ Our result shows that our function is not a one-to-one function. We get two separate answers joined by the keyword "or": $$a=b$$ $$\text{or}$$ $$a=-b - 10$$

#### 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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