Composite and inverse functions

Functions f and g are defined by

Using set notation, state the range of f

Determine whether f has an inverse.

Fully justify your answer.

The graph of y = g (x), and the line with equation y = x, are shown on the diagram below.

Sketch the graph of y = g^–1(x) on the diagram.

The composite function gf is denoted by h

Write down an expression for h (x)

State the range of h

The function f is defined by f(x) = ex + 1 for x ℝ

Find an expression for f–1(x)

Tick () one box.

f–1(x) = ln (x – 1)

f–1(x) = ln (x) – 1

The function f is defined by

The function g is defined by

g (x) = ln x

where g has its greatest possible domain.

Using set notation, state the range of f

State the domain of g

The composite function h is given by

Write down an expression for h(x) in terms of x

Determine if h has an inverse.

Fully justify your answer.

The function f is defined by

The curve with equation y f (x) is shown below.

State the value of a

On the diagram above, sketch the curve with equation

and

sketch the line with equation

Explain why the solution to the equation

must also be a solution to the equation

Use the Newton-Raphson method with x₀ 0 to find an approximate solution, x₃, to the equation

Give your answer to four decimal places.

The functions f and g are defined by

The function h has maximum possible domain and is defined by

Find an expression for h (x)

Find the domain of h

The function f is defined by

Solve the equation

Circle your answer.

The function h is defined by

where h has its maximum possible domain.

Find the domain of h.

Give your answer using set notation.

Alice correctly calculates

She then argues that since there is a change of sign there must be a value of x in the interval 1 x 4 that gives h(x) 0

Explain the error in Alice’s argument.

The function f is defined by

Find the range of f.

f ¹ x 6 (b) (i) Find ()

State the range of f ¹(x)

State the transformation which maps the graph of y f (x) onto the graph of

Find the coordinates of the point of intersection of the graphs of y f (x) and

Each of these functions has domain x

Which function does not have an inverse?

Circle your answer.

3 2 x

The function f is defined by f (x) e^x4, x

Find f ¹(x) and state its domain.