Determining a Function Algebraically

​Determining if an equation represents a function:

1. Solve for 𝒚 (if needed)

2. Determine how many outputs (𝒚) there are for a given input (𝒙).

*Remember, an equation is only a function if there is one, and only one, output for a given input.

​Example #1: Is 𝑦 −1=𝑥 a function?

1. Solve for y (if needed)

   y=x+1

2. Determine how many outputs (y) there are for a given input (x).

​If you substitute a value in for x, you will get only one value out for y. So, yes, 𝑦 −1=𝑥 is a function.

​Example #2: Is 𝑥+1= 𝑦² a function?

1. Solve for y (if needed)

   y=±√(x+1)

2. Determine how many outputs (y) there are for a given input (x).

​If you substitute a value in for x, you will get TWO values for y. For example, 𝑥=1, 𝑦= ±1+1=±2. Therefore, x+1= y² is NOT a function.

Multiple Choice

Is 3y=x+12 a function?

• Yes

• No

Multiple Choice

Is y³ = 4x-7 a function?

• Yes

• No

​Notice when y is raised to an odd exponent, the result is still a function because we don’t have the plus/minus result that we get from taking an even root.

If y is raised to an even exponent, the result will not be a function.

Multiple Choice

Is 5x=y⁶ a function?

• Yes

• No

Multiple Choice

Is 5x=y⁵ a linear function?

• Yes

• No

Multiple Choice

Does the following equation represent a LINEAR function? If so, what is its slope?

$$4y=8x-12$$

• No, the equation does not represent a function.

• The equation represents a function, but it is not a linear function.

• The equation represents a linear function, and the slope is 8.

• The equation represents a linear function, and the slope is 2.

Multiple Choice

Does the following equation represent a LINEAR function? If so, what is its slope?

$$4x^2+y^2=8$$

• No, the equation does not represent a function.

• The equation represents a function, but it is not a linear function.

• The equation represents a linear function, and the slope is 4.

• The equation represents a linear function, and the slope is 2.

Multiple Choice

Does the following equation represent a LINEAR function? If so, what is its slope?

$$y=7$$

• No, the equation does not represent a function.

• The equation represents a function, but it is not a linear function.

• The equation represents a linear function, and the slope is 0.

• The equation represents a linear function, and the slope is undefined.

Time for the QZ!

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