### About Vertical Line Test:

A relation is any set of ordered pairs (x,y). A function is a special type of relation where there is a one to one correspondence. Each first component or x-value corresponds to or is linked to exactly one second component or y-value. Many times we hear this read as ‘for each x, there can be only one y’. When we have a function, no vertical line will intersect the graph in more than one location.

Test Objectives

- Understand the definition of a relation
- Understand the difference between domain and range
- Demonstrate the ability to use the vertical line test to determine if a relation represents a function

#1:

Instructions: Determine if each relation is a function using the vertical line test, then list the domain and range.

a) $$y=-4x + 2$$

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#2:

Instructions: Determine if each relation is a function using the vertical line test, then list the domain and range.

a) $$y=x^2 + 7$$

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#3:

Instructions: Determine if each relation is a function using the vertical line test, then list the domain and range.

a) $$y=\frac{5}{x^2 - 1}$$

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#4:

a) $$y^2+x^2=16$$

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#5:

a) $$x=-3(y - 1)^2 - 1$$

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Written Solutions:

#1:

Solutions:

a) Yes - this relation is a function

$$domain=\left\{x|x ∈ ℝ\right\}$$ $$range=\left\{y|y ∈ ℝ\right\}$$

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#2:

Solutions:

a) Yes - this relation is a function

$$domain=\left\{x|x ∈ ℝ\right\}$$ $$range=\left\{y|y ≥ 7\right\}$$

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#3:

Solutions:

a) Yes - this relation is a function

$$domain=\left\{x|x ≠-1,1\right\}$$ $$range=\left\{y|y ≤-5~or~y > 0\right\}$$

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#4:

Solutions:

a) No - this relation is not a function

$$domain=\left\{x|-4 ≤ x ≤ 4\right\}$$ $$range=\left\{y|-4 ≤ y ≤ 4\right\}$$

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#5:

Solutions:

a) No - this relation is not a function

$$domain=\left\{x|x ≤ -1\right\}$$ $$range=\left\{y|y ∈ ℝ\right\}$$