Function Concepts • Topic 5 of 5

Function Interpretation

Interpreting a function means explaining what its inputs, outputs and parts mean in a real context. In a model like C(t) = 8t + 50 for the cost of a service after t hours, the input t is hours, the output C(t) is total cost, the slope 8 is the cost per hour, and the constant 50 is a fixed starting fee. A statement like C(3) = 74 means “after 3 hours the cost is $74.” The SAT often asks you to match a number to its meaning, or to read the rate and starting value from a model. Focus on what each variable and constant represents in the situation.

First-quadrant graph of C(t) = 8t + 50: the y-intercept is 50 and a slope triangle shows the cost rises 8 for each 1 hour.Interpreting a functiontC50+1+8Slope 8 = cost per hour; intercept 50 = fixed start

✅ Solved examples

1. For C(t) = 8t + 50, what does the 8 represent?
The cost per hour (rate of change).
2. For C(t) = 8t + 50, what does 50 represent?
The fixed starting cost when t = 0.
3. A tank holds W(t) = 5t + 20 liters after t minutes. What does W(4) = 40 mean?
After 4 minutes the tank holds 40 liters.
4. In P(n) = 12n, what does the 12 mean if n is items sold?
The price (revenue) per item.

✏️ Practice — try these, take hints as needed

1. For C(t) = 6t + 30, what does the 6 represent?
Coefficient of t = rate.
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The cost per hour.
2. For C(t) = 6t + 30, what does 30 represent?
Value at t = 0.
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The fixed starting cost.
3. A tank holds W(t) = 4t + 10 liters after t minutes. What does W(5) = 30 mean?
Input → output meaning.
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After 5 minutes the tank holds 30 liters.
4. In D(t) = 60t (distance in miles, t in hours), what does 60 mean?
Rate of change.
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The speed, 60 miles per hour.
5. In f(x) = 9x + 100, what is the value of the function when x = 0?
Set x = 0.
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100.

📝 Topic test — 8 questions

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