Functions and Relations Worksheet featuring graph analysis and function evaluation problems.

Let’s go through each problem one by one, carefully and clearly.

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Problems 1–3: Decide whether the graph represents y as a function of x. Explain your reasoning.

We use the vertical line test: If any vertical line crosses the graph more than once, it’s NOT a function. If every vertical line crosses at most once, it IS a function.

✔ Problem 1: The graph is a straight diagonal line going up to the right. No matter where you draw a vertical line, it will only hit the graph once.
→ Yes, this is a function.

✔ Problem 2: This is a smooth wave-like curve (like a sine or cosine shape). Again, no vertical line will ever cross it more than once.
→ Yes, this is a function.

✘ Problem 3: This graph looks like a sideways parabola — it opens to the right. If you draw a vertical line at x = 2, for example, it will cross the graph in TWO places (one above and one below the x-axis). That means one x-value gives two y-values → not allowed in a function.
→ No, this is NOT a function.

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Problems 4–6: Decide whether the relation is a function. If it is a function, give the domain and range.

A function means each input (x) has ONLY ONE output (y).

✔ Problem 4:
Input 0 → outputs 6 and -6 → ✘ Not a function (one input, two outputs)
Input 1 → outputs 5 and -5 → also not a function
→ Not a function.
(Domain = {0, 1}, Range = {6, -6, 5, -5} — but since it’s not a function, we don’t need to report domain/range per instructions.)

✔ Problem 5:
Inputs: 1, 2, 3, 4 → all map to output 4.
Each input has only ONE output → ✔ This IS a function.
Domain = {1, 2, 3, 4}
Range = {4}

✔ Problem 6:
Input 0 → output 6
Input 2 → output 6
Input 4 → output 3
Each input has only one output → ✔ This IS a function.
Domain = {0, 2, 4}
Range = {6, 3}

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Problems 7–12: Evaluate the function when x = 3, x = 0, and x = -2.

We plug each x-value into the function and calculate.

✔ Problem 7: f(x) = x
f(3) = 3
f(0) = 0
f(-2) = -2

✔ Problem 8: h(x) = x + 7
h(3) = 3 + 7 = 10
h(0) = 0 + 7 = 7
h(-2) = -2 + 7 = 5

✔ Problem 9: g(x) = x - 2
g(3) = 3 - 2 = 1
g(0) = 0 - 2 = -2
g(-2) = -2 - 2 = -4

✔ Problem 10: g(x) = 3x
g(3) = 3 × 3 = 9
g(0) = 3 × 0 = 0
g(-2) = 3 × (-2) = -6

✔ Problem 11: g(x) = 4x - 1
g(3) = 4×3 - 1 = 12 - 1 = 11
g(0) = 4×0 - 1 = 0 - 1 = -1
g(-2) = 4×(-2) - 1 = -8 - 1 = -9

✔ Problem 12: h(x) = 1.2x
h(3) = 1.2 × 3 = 3.6
h(0) = 1.2 × 0 = 0
h(-2) = 1.2 × (-2) = -2.4

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Final Answer:
1. Yes, it is a function (passes vertical line test).
2. Yes, it is a function (passes vertical line test).
3. No, it is not a function (fails vertical line test).
4. Not a function.
5. Function; Domain: {1, 2, 3, 4}, Range: {4}.
6. Function; Domain: {0, 2, 4}, Range: {6, 3}.
7. f(3)=3, f(0)=0, f(-2)=-2
8. h(3)=10, h(0)=7, h(-2)=5
9. g(3)=1, g(0)=-2, g(-2)=-4
10. g(3)=9, g(0)=0, g(-2)=-6
11. g(3)=11, g(0)=-1, g(-2)=-9
12. h(3)=3.6, h(0)=0, h(-2)=-2.4