Functions, inverse and composite functions

f(x) = ax + b and g(x) = where a and b are positive integers.

Prove that fg(x) – af^–1(x) is always a multiple of 3

Here is a number machine.

Work out the output when the input is 4

Work out the input when the output is 19

Complete this number machine.

Complete this number machine so that y = 4 x + 5

Complete this number machine so that y = 3 x – 24

Complete this number machine so that y = x

Complete this number machine so that y = 4 x + 5

Complete this number machine so that y = 3 x – 24

Complete this number machine so that y = x

Here is a number machine.

Work out the output when the input is 10

Work out the input when the output is 17

Match each equation to a statement to show what happens when the value of x is doubled. One has been done for you.

Here is a number machine.

Complete the number machine.

Here is a different number machine.

Complete the number machine.

Here is a different number machine.

When x = 5 y = 13 and when x = 10 y = 28

Complete the number machine.

f(x) = x – 3 g(x) = 4 x – 7

Work out the value of fg(6)

Solve (f(x))² = g(x)

g(x) = a × b where a and b are constants. g(0) = 8 and g(3) = 343

Work out the value of g(1)

Here are two number machines.

The outputs are the same.

Work out the value of a.

Here is a number machine.

Show that when the input increases by 2 the output increases by 2 a.

² f(x) = kx where k is a constant. Kai says that is equal to f(3) because = 3 Is he correct? Show working to support your answer.

f(x) = x² + 6x g(x) = 2 x + 4

Show that fg(x) = 4x² + 28x + 40

Solve fg(x) = – 5

Show that gf(2) is an integer.

Show that f^–1(8) is not an integer.

Luke wants to make a number machine so that y = 5 x + 3 Here is his attempt.

What mistake has he made?

f(x) = 2 x + 5 Show that 3f(x) – 12f^–1(x) simplifies to an integer.

Here is a number machine.

Work out the output.

Here is a different number machine.

Work out a formula for d in terms of c.

f(x) = 3x and g(x) = x² Circle the expression for fg(x)

f(x) = cx + d f(4) = 7 f(10) = 22 Work out the values of c and d.

24 (b) g(x) = 2x and h(x) =

Circle the expression for hg(x)

Here is a number machine.

Work out the output when the input is 16

Here is a different number machine.

Work out the output when the input is – 48

Complete this number machine.

Solve f –1 (x) = gf(x)

f(x) = 3x² – 4x + 8 for all values of x Jenny says, “f(10) must equal 2 × f(5), because 10 is 2 × 5” Is Jenny correct? Show working to support your answer.

f(x) = 2x – 3 and g(x) = x²

Work out the value of f^–1(3) + f(–0.5)

Complete the number machine.

Write down the output y in terms of x.

g(x) = 16 – x h(x) = x³ Solve gh(x) = 24

x =

Work out fg(x) Give your answer in the form a + bxⁿ where a, b and n are integers.

Work out f^–1(x)

f(x) = 5 – x and g(x) = 3 x + 7

Simplify f(2 x) + g(x – 1)

Solve g^–1(x) = 2x

x =

Complete the number machine so that q = 7 r – 2

Write down the output y in terms of x.

f(x) = x² – x³ Circle the value of f(–3)

18 –18 36 –36

g(x) = 6x² + 3 for all values of x. Work out fg(x). Give your answer in the form ax² + b where a and b are integers.

27

On the grid, draw the graph of the inverse function y = h^–1(x) for –2 x 2

For all values of x f(x) = sin x g(x) = x + 90 On the grid, draw the graph of the composite function y = fg(x) for 0° x 360°

Here is a number machine.

Work out the output when the input is 4

Work out the output when the input is –4