Function notation, evaluation, composition of functions
High School Essentials • 9-12
Functions are the language of mathematics. The notation f(x) -- read "f of x" -- is far more than a label; it precisely communicates which function is being used, what the input is, and what the output will be. This lesson covers how to read, evaluate, and combine functions, laying the groundwork for all of higher mathematics.
When we write f(x) = 2x + 3, we are defining a rule named f that takes any input x and produces the output 2x + 3. The notation f(x) does not mean "f times x." It means "the output of function f when the input is x."
We can use any letter for the function name (f, g, h, etc.) and any variable for the input. The expression f(5) means "evaluate the function f at x = 5."
Given f(x) = x2 - 4x + 1, find f(3), f(-2), and f(a + 1).
Notice that for f(a + 1), you replace every x with the entire expression (a + 1), using parentheses to ensure correct evaluation.
The domain is the set of all valid inputs. The range is the set of all possible outputs.
| Function Type | Domain Restriction | Example |
|---|---|---|
| Polynomial | All real numbers | f(x) = x2 + 1 |
| Rational | Denominator cannot equal 0 | g(x) = 1/(x - 3), domain: x is not 3 |
| Square root | Expression under radical must be ≥ 0 | h(x) = sqrt(x - 2), domain: x ≥ 2 |
Find the domain of g(x) = sqrt(5 - x).
Composition chains two functions together: the output of one becomes the input of the other. The notation (f ∘ g)(x) or f(g(x)) means "first apply g to x, then apply f to the result."
Think of it as a two-step process: g processes the input first, producing g(x). Then f takes that result and produces f(g(x)).
Let f(x) = 3x + 1 and g(x) = x2 - 2. Find f(g(x)) and g(f(x)).
f(g(x)): Replace x in f with g(x):
g(f(x)): Replace x in g with f(x):
Notice that f(g(x)) is not equal to g(f(x)). Composition is not commutative -- order matters.
f(x) = x + 3 does not mean f * x = x + 3. The notation f(2) means "substitute 2 for x," giving 2 + 3 = 5. Similarly, f(a + b) means substitute (a + b) for x, not f * a + f * b.
When computing f(g(2)), work from the inside out. First find g(2), then plug that number into f. This avoids errors from trying to substitute entire expressions at once.
1. If f(x) = 2x - 7, find f(4) and f(-3).
f(4) = 2(4) - 7 = 8 - 7 = 1. f(-3) = 2(-3) - 7 = -6 - 7 = -13.
2. Find the domain of h(x) = 1 / (x2 - 9).
The denominator cannot equal zero: x2 - 9 = 0 gives x = 3 or x = -3. Domain: all real numbers except x = 3 and x = -3.
3. Let f(x) = x + 4 and g(x) = 2x. Find f(g(3)) and g(f(3)).
f(g(3)): first g(3) = 2(3) = 6. Then f(6) = 6 + 4 = 10.
g(f(3)): first f(3) = 3 + 4 = 7. Then g(7) = 2(7) = 14.
4. Let f(x) = x2 and g(x) = x - 5. Write a formula for f(g(x)) and g(f(x)).
f(g(x)) = (x - 5)2 = x2 - 10x + 25.
g(f(x)) = x2 - 5 = x2 - 5.
5. If f(x) = 3x + 2, find f(x + h) - f(x) and simplify. (This is related to the difference quotient.)
f(x + h) = 3(x + h) + 2 = 3x + 3h + 2. Then f(x + h) - f(x) = (3x + 3h + 2) - (3x + 2) = 3h.
Function notation f(x) precisely identifies a function and its input. Evaluating a function means substituting the input value for x everywhere it appears. Domain is the set of allowable inputs; range is the set of resulting outputs. Composition f(g(x)) applies g first, then f to the result -- and the order matters because composition is not commutative. These concepts are foundational for calculus, where functions and their combinations are analyzed in depth.