LCAT Math, Advanced Math
Functions for the LCAT: lesson and practice
f(a) means "substitute a for x". f(g(x)) means apply g first, then f. For transformations, changes inside the bracket move the graph horizontally in the opposite direction, and changes outside move it vertically in the same direction.
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No calculator on the LCAT. LUMS’s official guidelines say calculators are not allowed, so every practice question here can be solved by hand.
| Transformation | Effect on point (p, q) |
|---|---|
| f(x − 3) | right 3: (p + 3, q) |
| f(x) + 2 | up 2: (p, q + 2) |
| −f(x) | reflect over x-axis: (p, −q) |
| f(−x) | reflect over y-axis: (−p, q) |
Composition
f(x) = 5x + 8, g(x) = 2x + 4
g(2) = 8, then f(8) = 48
g(2) = 8, then f(8) = 48
Common mistakes
Treating f(a) + f(b) as f(a + b)Evaluate each separately.
Composing in the wrong orderf(g(x)): inside first.
Shifting the wrong wayf(x − 3) moves right, not left.
Sample questions with answers
1. If f(x) = 7x − 8, what is f(5) + f(−2)?
- 13
- 5 (correct)
- 27
- −594
f(5) = 27 and f(−2) = −22; the sum is 5.
2. (−5, 5) is on y = f(x). Which point is on y = f(x + 4) + 3?
- (−1, 8)
- (−9, 2)
- (−9, 8) (correct)
- (−2, 1)
x + 4 moves left 4; + 3 moves up 3.
3. If f(x) = 5x + 8 and g(x) = 2x + 4, what is f(g(2))?
- 40
- 48 (correct)
- 144
- 8
g(2) = 8 and f(8) = 48.
Frequently asked questions
What does f(x − h) do to a graph?
It shifts the graph h units to the right.
How do I evaluate f(g(x))?
Find g(x) first, then substitute the result into f.
Is functions tested on the LCAT?
Yes. It falls under Advanced Math, one of the content domains LUMS lists for the LCAT in its official sample questions.
Can I practice this without a calculator?
Yes. Every practice question here can be solved by hand, because calculators are not allowed on the LCAT.