Functions (Cambridge International AS Mathematics): Exam Questions

State whether each of the following mappings is one-one, many-one, one-many or many-many.

(i) \text{f}: x \mapsto 4x - 2

(ii) \text{f}: x \mapsto x^{2}

(iii) \text{f}: x \mapsto \dfrac{x}{4}

(iv) \text{f}: x \mapsto \sqrt{x}

The function $\text{f}$ is defined by

$\text{f} ( x ) = x^{2} - 8 x - 20$

for $x \in ℝ$.

Sketch the graph of $y = \text{f} ( x )$, giving the coordinates of the points where the graph meets the coordinate axes.

Given that the minimum point of the graph has $x$-coordinate $4$, state the range of $\text{f}$.

The function $\text{f}$ is defined by

$\text{f} ( x ) = x^{2} - 9$

for $x \geq 3$.

(i) State the range of $\text{f}$.

(ii) The domain of $\text{f}$ is changed to $x \geq 0$. State the range of $\text{f}$ for this new domain.

The functions $\text{f}$ and $\text{g}$ are defined by

$\text{f} ( x ) = 3 x + 5$

$\text{g} ( x ) = - 2 x$

for $x \in ℝ$.

Find an expression for

(i) $\text{fg} ( x )$,

(ii) $\text{gf} ( x )$.

Solve the equation $\text{f} ( x ) = \text{g} ( x )$.

State whether each of the following mappings is one-one, many-one, one-many or many-many.

(i) \text{f}: x \mapsto x^{2}

(ii) \text{f}: x \mapsto 3x + 1

(iii) \text{f}: x \mapsto (x + 1)^{3}

(iv) \text{f}: x \mapsto \pm\sqrt{x}

The function $\text{f}$ is defined by

$\text{f} ( x ) = x^{2} - 4$

for $x \geq 0$.

State the range of $\text{f}$.

The domain of $\text{f}$ is changed to $x \leq 0$. State the range of $\text{f}$ for this new domain.

The function $\text{f}$ is defined by

$\text{f} ( x ) = 3 x^{2} + 1$

for $x \geq 0$.

Find an expression for $\text{f}^{-1} ( x )$.

State the domain and range of $\text{f}^{-1}$.

The function $\text{f}$ is defined by

$\text{f} ( x ) = x^{2} + 2 x - 3$

for $x \in ℝ$.

Sketch the graph of $y = \text{f} ( x )$, giving the coordinates of the points where the graph meets the coordinate axes and the coordinates of the turning point.

Write down the range of $\text{f}$.

The functions $\text{f}$ and $\text{g}$ are defined by

$\text{f} ( x ) = x^{2}$

$\text{g} ( x ) = 4 x - 3$

for $x \in ℝ$.

Write down the range of $\text{f}$.

Find an expression for

(i) $\text{fg} ( x )$,

(ii) $\text{gf} ( x )$.

Solve the equation $\text{f} ( x ) = \text{g} ( x )$.

The diagram shows the graph of $y = \text{f} ( x )$ (solid line) together with a dotted line.

Use the graph to write down the domain and range of $\text{f}$.

Given that $( 1 , 1 )$ lies on the dotted line, write down the equation of the dotted line.

On the diagram, sketch the graph of $y = \text{f}^{-1} ( x )$.

The function $\text{f}$ is defined by

\text{f}: x \mapsto \dfrac{x^{2} + 1}{x^{2}}

for $x \in ℝ , x \neq 0$.

Show that $\text{f} ( x )$ can be written as $1 + \frac{1}{x^{2}}$.

Explain why $\text{f}$ does not have an inverse, and suggest an adaptation to its domain so that the inverse exists.

The domain of $\text{f}$ is changed to $x > 0$. Find an expression for $\text{f}^{-1} ( x )$, and state its domain and range.

The functions $\text{f}$ and $\text{g}$ are defined by

$\text{f} ( x ) = \frac{1}{2} ( 4 x - 3 )$

$\text{g} ( x ) = 0 . 5 x + 0 . 75$

for $x \in ℝ$.

Find an expression for

(i) $\text{fg} ( x )$,

(ii) $\text{gf} ( x )$.

Write down $\text{f}^{-1} ( x )$ and state its domain and range.

The function $\text{f}$ is defined by

$\text{f} ( x ) = \frac{2}{x}$

Write down the domain of $\text{f}$.

Sketch the graph of $y = \text{f} ( x )$, stating the coordinates of any intersections with the axes and the equations of any asymptotes.

Write down the range of $\text{f}$.

The functions $\text{f}$ and $\text{g}$ are defined by

$\text{f} ( x ) = 3 x^{2} + 2$

$\text{g} ( x ) = 1 - 3 x$

for $x \in ℝ$.

Write down the range of $\text{f}$.

Find an expression for

(i) $\text{fg} ( x )$,

(ii) $\text{gf} ( x )$.

Solve the equation $\text{f} ( x ) = \text{g} ( x ) + 1$.

The graph of y = f \left(x\right) is shown below.

The diagram shows the graph of $y = \text{f} ( x )$ (solid curve) together with a dotted line.

(i) Use the graph to write down the domain and range of $\text{f}$.

(ii) Write down the equation of the dotted line.

On the diagram, sketch the graph of $y = \text{f}^{-1} ( x )$.

The function $\text{f}$ is defined by

$\text{f} ( x ) = 4 x^{3} + 4 x^{2} - 7 x + 2$

Write down the domain and range of $\text{f}$.

Sketch the graph of $y = \text{f} ( x )$, stating the coordinates of any intersections with the axes. (You do not need to give the coordinates of any turning points.)

The functions $\text{f}$ and $\text{g}$ are defined by

$\text{f} ( x ) = x^{2} - 2$ for $x \in ℝ$,

$\text{g} ( x ) = 1 - \frac{2}{x}$ for $x \in ℝ , x \neq 0$.

Write down the range of $\text{f}$.

Leaving your answers as single fractions, find an expression for

(i) $\text{fg} ( x )$,

(ii) $\text{gf} ( x )$.

Given that $( x - 1 )^{2}$ is a factor of $x^{3}-3x+2$, solve the equation $\text{f} ( x ) = \text{g} ( x )$.

The function $\text{f}$ is defined by

\text{f}: x \mapsto \sqrt{25 - x^{2}}

for $x \in ℝ , - 5 \leq x \leq 5$.

Explain why $\text{f}$ does not have an inverse.

Suggest an adaptation to the domain of $\text{f}$ so that:

• the inverse of $\text{f}$ exists,
• the graph of $y = \text{f} ( x )$ lies in the first quadrant only, and
• the domain of $\text{f}$ is as large as possible.

State the range of your adapted $\text{f}$.

The domain of $\text{f}$ is changed to $- 5 \leq x \leq 0$. Find an expression for $\text{f}^{-1} ( x )$, and state its domain and range.

The function $\text{f}$ is defined by

$\text{f} ( x ) = ( x - 3 )^{2} ( x - 4 )^{2}$

for $2 \leq x \leq 5$.

Work out the range of $\text{f}$.

If the domain of $\text{f}$ is changed to $x \leq 2$, what is the range of $\text{f}$?

State another domain for $\text{f}$ that would have the same effect as that in part (b).

The functions $\text{f}$ and $\text{g}$ are defined by

$\text{f} ( x ) = ( x - 1 )^{2} - 4$ for $x \geq 1$,

$\text{g} ( x ) = 1 + \sqrt{x+4}$ for $x \geq - 4$.

Find an expression for

(i) $\text{fg} ( x )$,

(ii) $\text{gf} ( x )$.

Write down $\text{f}^{-1} ( x )$ and state its domain and range.

The graphs of $\text{f}$ and $\text{f}^{-1}$ are drawn on the same axes. Describe the transformation that maps one graph onto the other.

Find the coordinates of the point where the graphs of $y = \text{f} ( x )$ and $y = \text{f}^{-1} ( x )$ meet.

The diagram shows the graph of $y = \text{f} ( x )$ together with the line $y = x$ (dotted). The function $\text{f}$ has rotational symmetry about the origin, and for $x > 0$ there is a vertical line of symmetry at $x = 4 . 5$.

Use the graph to write down the domain and range of $\text{f}$.

On the diagram, sketch the reflection of $\text{f}$ in the line $y = x$, and explain why this cannot be the graph of $\text{f}^{-1} ( x )$.

(i) Given that the maximum solution to $\text{f} ( x ) = 6$ is $x = 6$, state the restriction on the domain of $\text{f}$ such that $\text{f}^{-1}$ exists.

(ii) Hence write down the domain and range of $\text{f}^{-1}$.