Preparing for University Calculus - Math and Computer Science

3.4 FunctionsYou should understand the concept of “function” and “inverse function” and knowhow to compute the composition of two or more functions. You should be able todetermine the range and domain of a simple function. This will be important forunderstanding the Chain Rule, various methods of integration, and limits.Examples:1. Find the domain of the functions defined by:(a) f(x) = 1x 2 −1(b) g(x) = √ 4 − xSolution(a) The domain of f(x) = 1x 2 −1 is the set of all values of x such that x2 −1 ≠0;that is x ≠ ±1 orx ∈{(−∞, −1) ∪ (−1, 1) ∪ (1, ∞)}(b) The domain of g(x) = √ 4 − x is the set of all values of x such that4 − x ≥ 0; that is x ≤ 4orx ∈ (−∞, 4].2. If f(x) =3x + 5, findSolutionf(x + h) − f(x)hf(x + h) − f(x)h== 3h h[3(x + h)+5]− [3x +5]h= 3=3x +3h +5− 3x − 5h3. If f(x) = √ x − 1 and g(x) =x 2 , find:(a) f(g(x)) (b) g(f(x))Solution(a) f(g(x)) = f(x 2 )= √ x 2 − 1; this has domain (−∞, −1] ∪ [1, ∞).(b) g(f(x) =g( √ x − 1) = ( √ x − 1) 2This is x − 1 on the domain [1, ∞) but is undefined elsewhere.16