Polynomial, rational and composite functions

A polynomial function has the form f(x) = aₙxⁿ + … + a₁x + a₀. A rational function is a ratio of polynomials, such as f(x) = (x + 1)/(x − 2); it is undefined where the denominator is zero, giving a vertical asymptote (x = 2 here). The composite function f∘g means "do g first, then f": (f∘g)(x) = f(g(x)).

Worked exampleComposite functions

f(x) = 2x + 1 and g(x) = x². Find f(g(x)) and g(f(x)).

f(g(x)) replaces x in f by x².
g(f(x)) squares f.

Answer: The two are different: composition is not commutative.

The graph approaches the vertical asymptote x = 2 and the horizontal asymptote y = 1.

Exponential and logarithmic functions

An exponential function y = aˣ (a > 0, a ≠ 1) passes through (0, 1) and has the x-axis as an asymptote. Its inverse is the logarithmic function y = logₐ x, which passes through (1, 0). The most useful base is e ≈ 2.718: y = eˣ and y = ln x. The two graphs are mirror images in y = x.

Laws of logarithms.

Always positive and increasing; it grows faster than any power.

The reflection of eˣ in y = x. It is defined only for x > 0.

Try itComposite function

Choose f(x) = ax + b and g(x) = x². Evaluate f(g(x)) and g(f(x)) at x.

Check yourself If f(x) = x + 3 and g(x) = 2x, then f(g(4)) =

Check yourself The inverse of y = eˣ is

Check yourself The vertical asymptote of (x + 1)/(x − 2) is

Key terms

Composite function: A function applied to the result of another.
Asymptote: A line that a curve approaches without meeting.
Exponential function: y = aˣ.
Logarithmic function: The inverse of an exponential function.

Summary

f(g(x)) does g first; composition is not commutative.
Rational functions have asymptotes where the denominator is zero.
Exponential and logarithmic functions are inverses.
ln x is defined only for x > 0.