Functions: Composite Functions

1. The functions f and g are defined by f(x) = x² + 1 and g(x) = 3x – 4, on the set of real numbers. Find g(f(x)). (2)

2. Functions f and g are defined on the set of real numbers by
f(x) = x² + 3
g(x) = x + 4

Find expressions for:

(a) f(g(x)) (2)

(b) g(f(x)) (1)

3. Functions f and g are defined on suitable domains by
f(x) = x(x – 1) + q, and
g(x) = x + 3

Find an expression for f(g(x)). (2)

4. Functions f and g are defined on suitable domains by f(x) = cosx and g(x) = x + π6

$\frac{\pi }{6}$

. What is the value of f(g(π6

$\frac{\pi }{6}$

))? (3)

5. Functions f, g and h are defined on the set of real numbers by
f(x) = x³ – 1,
g(x) = 3x + 1,
h(x) = 4x – 5

(a) Find g(f(x)) (2)

(b) Show that g(f(x)) + xh(x) = 3x³ + 4x² – 5x – 2. (1)

6. The functions f and g are defined by
f(x) = 4x + 3, x ∈ R, and
g(x) = 1x2

$\frac{1}{x^2}$

, x ∈ R, x ≠ 0.

Write down:

(a) the composite function f(g(x)) (1)

(b) the inverse function f^-1(x). (2)