Functions, inverse & composite functions

Here is a number machine.

Find the output when the input is 6

Find the input when the output is –11

17g(x) = 1 – 3xh(x) = 2x² – 1 Show that 3gh(x) + hg(x) = 0 has just one solution for x.

Here is a number machine.

Find the output when the input is 11

Here is a different number machine.

When the input is 10, the output is 28

Complete this number machine.

Find f (x)

For all values of x g(x) = (x – 1)²and h(x) = 1 – 2x

Work out the value of gh(5)

Here is a number machine.

Work out the output when the input is 13

Work out the input when the output is 28

Show that there is a number for which the output is the same as the input.

The functions f and g are given by

Find f(–3)

Find fg(1)

Find g^–1(4)

The functions f and g are such that

Find gf(−3)

Find g⁻¹(x)

g⁻¹(x) =_........

20For x 0, the functions f and g are such that

Find g^–1(x)

g^{– 1}(x) =_........

Solve gf (x) = 3

h(x) = fg(x) Find h⁻¹(x)

h⁻¹(x) =_........

The functions g and h are such that

Find g(16)

Find hg⁻¹(x) Give your answer in terms of x in its simplest form.

hg⁻¹(x) =_........

The functions f and g are such that

Work out gf(1)

The function h is such that h = (fg)^–1

Find h(x)

12The diagram shows a number machine.

Find the output when the input is 7

Find the input when the output is 41

f and g are functions such that

Find g(5)

Find gf(9)

Find g^–1(6)

The function f is given by

Show that f^–1(50) = 3

The functions g and h are given by

Find the values of x for which

The functions f and g are such that

Find f^–1(x)

Given that fg(x) = 2gf(x),

show that 15x² – 12x – 1 = 0

For all values of x

f(x) = (x + 1)² and g(x) = 2(x − 1)

Show that gf(x) = 2 x (x + 2)

Find g⁻¹(7)

Here is a number machine.

Work out the output when the input is 8

Work out the input when the output is 28

f(x) = 4sin x °

Find f(23) Give your answer correct to 3 significant figures.

g(x) = 2 x – 3

Find fg(34) Give your answer correct to 3 significant figures.

h(x) = (x + 4)²

Ivan needs to solve the following equation h(x) = 25

He writes

x + 4 = 5 x = 1

This is not fully correct.

Explain why.

f and g are functions such that

Find f(–5)

Find fg(1)

The functions f and g are such that

g(3) = 20 and f⁻¹

Find the value of a and the value of b.