Strategy #12: Interpret Function Notation
Category: Advanced Math
f(x) means "the output of function f when the input is x." To find f(3), substitute 3 everywhere you see x.
Why it matters: Function notation appears in straightforward evaluation problems but also in trickier forms: f(x + 1), f(f(x)), or g(f(x)) (composition).
Key skill — composition: If f(x) = 2x + 1 and g(x) = x², then g(f(x)) = g(2x + 1) = (2x + 1)².
Practice tip: For any function problem, identify the input clearly before substituting. With composition, always work from the inside out: evaluate the inner function first.
| x | g(x) | f(x) |
|---|---|---|
| −27 | 3 | |
| −9 | 0 | |
| 21 | 5 |
The table shows three values of x and their corresponding values of g(x), where g(x) = f(x)/(x + 3) and f is a linear function. What is the y-intercept of the graph of y = f(x) in the xy-plane?
A) (0, 36)
B) (0, 12)
C) (0, 4)
D) (0, −9)
Solution:
| x | g(x) | f(x) |
|---|---|---|
| −27 | 3 | −72 |
| −9 | 0 | 0 |
| 21 | 5 | 120 |
Since g(x) = f(x)/(x + 3), we get f(x) = g(x)(x + 3).
g(−27) = 3 = f(−27)/(−27 + 3) = f(−27)/−24 → f(−27) = 3(−24) = −72
g(−9) = 0 = f(−9)/(−9 + 3) = f(−9)/−6 → f(−9) = 0(−6) = 0
g(21) = 5 = f(21)/(21 + 3) = f(21)/24 → f(21) = 5(24) = 120
Compute the slope for f(x): m = (0 − (−72))/(−9 − (−27)) = 72/18 = 4
Now write its equation in point-slope form: y − 0 = 4(x + 9)
Distributing the 4, we see that the y-intercept is (0, 36).
The answer is A.