Functions (Edexcel IGCSE Maths): Exam Questions

Show that $\frac{x^{2}+3x}{2x^{2}+5x-3}$ can be written as $\frac{x}{kx-1}$

State the value of $k$.

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

Find the inverse function $f^{-1}$ in the form $\mathrm{f}^{- 1} ( x ) = .......$

Show your working clearly.

The functions $f$and $g$ are defined as

$\begin{gathered} \mathrm{f} ( x ) & = x^{2} + 6 \\ \mathrm{g} ( x ) & = x - 10 \end{gathered}$

Solve the equation $\mathrm{fg}(x)=f(x)$

Show clear algebraic working.

$f$ and $g$ are functions such that

f ( x ) = 2 / x ^2 and g ( x ) = 4 x ^3

Find $f(-5)$

Find $fg ( 1 )$

The functions $f$ and $g$ are such that

$f(x)=5x+3g(x)=ax+b$ where $a$ and $b$ are constants.

$g(3)=20\mathrm{and}f^{-1}(33)=g(1)$

Find the value of $a$ and the value of $b$.

$f(x)=3x^{2}---2x---8$

Express $f(x+2)$ in the form $ax^{2}+bx$

$\mathrm{f} ( x ) = 3 x - 2 \mathrm{g} ( x ) = \frac{ 10}{x + 2}$

Express the inverse function $f^{-1}$ in the form $\mathrm{f}^{- 1} ( x ) = ...$

Find $\mathrm{gf}(x)$ Simplify your answer.

$\mathrm{f} ( x ) = \sqrt{x - 6}$

Find f(10)

State which values of $x$ must be excluded from a domain of f

The diagram shows part of the graph of $y = \mathrm{g} ( x )$

Find $\mathrm{g} ( 2 )$

Find $fg ( 0 )$

One of the solutions of $\mathrm{g} ( x ) = k$, where $k$ is a number, is $x=1$

Find the other solutions Give your answers correct to 1 decimal place

The functions $f$ and $g$ are such that $f(x)=x+3$ and $\mathrm{g} ( x ) = \frac{1}{x - 2}$

Find $\mathrm{fg}(x)$ Give your answer as a single algebraic fraction expressed as simply as possible.

Express the inverse function $g^{-1}$ in the form $\mathrm{g}^{- 1} ( x ) = ...$

f is a function such that

$\mathrm{f} ( x ) = \frac{1}{x^{2} + 1}$

Find $\mathrm{f} ( 1 \frac{1}{2} )$

$g$ is a function such that

$\mathrm{g} ( x ) = \sqrt{x - 1} x greater-than \text{or} \text{slanted} \text{equal} \text{to} 1$

Find $\mathrm{f} g ( x )$

Give your answer as simply as possible.

The diagram shows the graph of $y=f(x)$ for negative 3.5 less-than or slanted equal to x less-than or slanted equal to 1.5

Use the graph of $\mathrm{f} ( x )$ to find the value of $\mathrm{f} ( 0 )$.

For which values of $k$ does the equation $\mathrm{f} ( x ) = k$ have only one solution?

Find an estimate for the gradient of the curve at the point where $x$ = −2.5

$\mathrm{g} ( x ) = \frac{1}{2 + x}$

State which value of $x$ must be excluded from any domain of $g$

Find fg(-3)

The function $f$ is such that

$f(x)=4x---1$

Find $f^{-1}(x)$

The function $g$ is such that

$g(x)=kx^{2}$ where $k$ is a constant.

Given that $\mathrm{fg}(2)=12$

work out the value of $k$

$f$is the function $f(x)=2x+5$

Find $f(3)$

Express the inverse function $f^{-1}$ in the form $\mathrm{f}^{- 1} ( x )$ =

$g$ is the function $g(x)=x^{2}---25$

Find $g(---3)$

(i) Find $\mathrm{gf}(x)$ Give your answer as simply as possible.

(ii) Solve $\mathrm{gf}(x)=0$

The function $g$ is defined as

$g : x \rightarrow \text{from} \text{bar} 5 + 6 x - x^{2}$ with domain { x colon x greater-than or slanted equal to 3 }

Express the inverse function g⁻¹ in the form $g^{- 1} : x \rightarrow \text{from} \text{bar} ...$

$g^{- 1} : x \rightarrow \text{from} \text{bar}$...................................................

State the domain of $g^{-1}$

The functions $f$ and $g$ are such that

f ( x ) = 3 x - 1 and g ( x ) = x ^2 + 4

Find $f^{-1}(x)$

Given that $\mathrm{fg}(x)=2\mathrm{gf}(x)$,

show that $15x^{2}-12x---1=0$

The functions $f$ and $g$ are defined as

$\mathrm{f} ( x ) = 1 \frac{1}{2} x + 4 \mathrm{g} ( x ) = \frac{2 x}{x + 1}$

Work out $f(6)$

Work out $\mathrm{fg}(---3)$

$g(a)=-2$

Work out the value of $a$.

Express the inverse function $f^{-1}$ in the form $f^{-1}(x)$ = ...

The functions $f$ and $g$ are such that

$f(x)=x+25g(x)=x^{2}--12x$

The function $h$ is such that $h(x)=\mathrm{fg}(x)$

The domain of $h$ is { x colon x less-than or slanted equal to 6 }

Express the inverse function $h^{--1}$ in the form $h^{--1}(x)=...$

$h^{--1}(x)=.....................$

The function f is defined as

$f(x)=\frac{x-6}{2}$

Find f (8)

Express the inverse function $f^{-1}$in the form $\mathrm{f}^{- 1} ( x )$ = ......

The function $g$ is defined as

$g(x)=\sqrt{x-4}$

Which values of $x$ cannot be included in a domain of $g$?

Express the function $\mathrm{gf}$ in the form $gf ( x )$= .......

Give your answer as simply as possible.

f is the function such that f(x) = 2x - 5

g is the function such that g(x) = x² - 10

Find f (4)

Find fg(-4)

Express the inverse function f^-1 in the form f^-1 (x) = ...

Solve gf (x) = -1

The functions $f$ and $g$ are defined as

$\begin{gathered} \mathrm{f} ( x ) & = 5 x^{2} - 10 x + 7 \text{where} x greater-than \text{or} \text{slanted} \text{equal} \text{to} 1 \\ \mathrm{g} ( x ) & = 7 x - 6 \end{gathered}$

Find $\mathrm{fg}(2)$

Express the inverse function $f^{-1}$ in the form $f^{-1}(x)=...$

$f^{-1}(x)=$...............................

$\mathrm{f} ( x ) = \frac{3}{x + 1} + \frac{1}{x - 2}$

State one value of $x$ which cannot be included in any domain of f.

Find the value of f(0)

Find the value of $x$ for which $\mathrm{f} ( x ) = 0$

Show clear algebraic working.

The functions $f$ and $g$ are such that

$f(x)=3(x---4)\mathrm{and}g(x)=\frac{x}{5}+1$

Find the value of $\mathrm{f} ( 10 )$

Find $\mathrm{g}^{- 1} ( x )$

Show that $\mathrm{ff}(x)=9x---48$

The functions $f$ and $g$are such that

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

The function $h$ is such that $\mathrm{h} ( x ) = \text{fg} ( x ) \text{for} x greater-than \text{or} \text{slanted} \text{equal} \text{to} - 2$

Express the inverse function $h^{-1}(x)$ in the form $h^{-1}(x)$ = .........

$h^{-1}(x)$ = ................

The diagram shows parts of the graphs of $y = \mathrm{f} ( x )$and $y = g ( x )$.

Find g(0)

Find $g$f(–1)

The function f is defined as f($x$) = $\frac{3}{4+x}$

Find the value of f(1)

State which value of $x$ must be excluded from any domain of f.

The function $g$ is defined as $g(x)$ = 5 + $x$

Given that $g(a)$ = 7, find the value of $a$.

Calculate $fg$(1)

Find $fg ( x )$ Simplify your answer.

The function$f$ is such that $f(x)=5+6x--x^{2}$ for $x \text{less} - \text{than} or \text{slanted} \text{equal} to 3$

Express $5+6x--x^{2}$ in the form $p--(x--q)^{2}$ where $p$ and $q$ are constants.

Using your answer to part (a), find the range of values of $x$ for which $f^{-1}(x)$ is positive.

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

$\mathrm{g} ( x ) = \frac{x + 1}{x}$

State which value of $x$cannot be included in the domain of f or $g$.

Solve $gf ( a ) = 3$

Express the inverse function $g^{-1}$ in the form $\mathrm{g}^{- 1} ( x )$

The function $h$ is defined as

$\mathrm{h} ( x ) = \frac{2 x - 4}{x}$

State the value of $x$ that cannot be included in the domain of h

Express the inverse function h^–1 in the form $h^{--1}(x)=...$

$h^{--1}(x)=$....................

$g$ is the function with domain $x \text{greater} - \text{than} or \text{slanted} \text{equal} to - 3$ such that $g(x)=x^{2}+6x$

Write down the range of $g^{-1}$

Express the inverse function $g^{-1}$ in the form g ^ - 1 colon x -> from bar horizontal ellipsis

$g^{- 1} : x \rightarrow \text{from} \text{bar}$

f:$x$ → 2$x^{2}\quad +$1 $g$:$x$→ $\frac{2x}{x-1}$ where $x$≠ 1

Express the composite function gf in the form $g$f:$x$ → .....

Give your answer as simply as possible.

Express the inverse function $g^{-1}$in the form $\mathrm{g}^{- 1} ∶ x$→ ......

The function $f$ is such that $f(x)=(x-4)^{2}$ for all values of $x$.

Find $f(1)$

State the range of the function $f$

The function $g$ is such that $\mathrm{g} ( x ) = \frac{4}{x + 3} x \text{eq} - 3$

Work out $\mathrm{fg}(2)$

The function f is defined as $f(x)=\frac{\sqrt{x^{2}+k^{2}}}{x}$ for $x>0$ and where $k$ is a positive number.

Find the value of $p$ for which $f^{-1}(p)=k$

$p=$

The function $g$ is defined as $g(x)=x^{2}\mathrm{for}x>0$

Given that $\mathrm{gf}(a)=k\mathrm{for}k>1$ find an expression for $a$ in terms of $k$.

$a=$

$\mathrm{g} ( x ) = 3 x + 7$

Solve $g^{--1}(x)=2x$

$x$ = ....................

The functions $g$ and h are defined as

$\mathrm{g} ( x ) = \frac{x}{2 x - 5}$

$\mathrm{h} ( x ) = x + 4$

Find the value of $g$(1)

State which value of $x$ must be excluded from any domain of $g$

Find $gh ( x )$

Simplify your answer

Express the inverse function $g^{-1}$ in the form $\mathrm{g}^{- 1} ( x )$ = .....

$f(x)=3x$

Circle the expression for$f^{-1}(x)$

$\mathrm{f} ( x ) = 1 \frac{1}{2} x$ $\mathrm{g} ( x ) = x - x^{2}$

Solve $f^{--1}(x)=\mathrm{gf}(x)$

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

Find the value of $\mathrm{f} ( 11 )$

State which value of $x$must be excluded from any domain of f

Find $\mathrm{f}^{- 1} ( x )$

State the value which cannot be in any range of f

A function machine is shown below.

If the Input is 3, the Output is 5.

If the Input is 7, the Output is 25.

Use this information to fill in the two boxes.

A function is represented by the following function machine.

A number is input into the machine. The output is used as a new input. The second output is 11.

Work out the number that was the first input.

A number is input into the machine. The output given is the same number.

Work out the number.

$\mathrm{f} ( x ) = \frac{2 x}{5} - 1$

Work out the value of $f^{--1}(3)+f(--0.5)$

f is the function such that

$\mathrm{f} ( x ) = \frac{x}{3 x + 1}$

Find f(0.5)

Find ff(−1)

Find the value of $x$ that cannot be included in any domain of f

Express the inverse function $f^{-1}$ in the form $\mathrm{f}^{- 1} ( x )$= ......

Show clear algebraic working.

$\begin{array}{rcl} \mathrm{f} ( x ) & = c x + d \\ \mathrm{f} ( 4 ) & = 7 \\ \mathrm{f} ( 10 ) & = 22 \end{array}$

Work out the values of $c$ and $d$.

$c=$.........................

$d=$.........................

$\mathrm{g} ( x ) = 2 x$ and $\mathrm{h} ( x ) = \frac{x - 1}{2}$

Circle the expression for $\mathrm{hg}(x)$

The functions $g$ and $h$ are such that

$\mathrm{g} ( x ) = \frac{11}{2 x - 5}$

$\mathrm{h} ( x ) = x^{2} + 4 x greater-than \text{or} \text{slanted} \text{equal} \text{to} 0$

What value of $x$ must be excluded from any domain of $g$?

Solve $\mathrm{gh}(x)=1$

$\mathrm{f} ( x ) = 5 - x$ and $\mathrm{g} ( x ) = 3 x + 7$

Simplify $f(2x)+g(x--1)$

The function f is such that $f(x)=x^{2}--8x+5$ where $x \text{less} - \text{than} or \text{slanted} \text{equal} to 4$ Express the inverse function $f^{-1}$ in the form $f^{--1}(x)=...$

$f^{--1}(x)=................$

The functions $f$ and $g$ are such that

$\mathrm{f} : x \rightarrow \text{from} \text{bar} 5 x + 7 \mathrm{g} : x \rightarrow \text{from} \text{bar} \frac{5}{2 x - 9}$

State which value of $x$ cannot be included in any domain of $g$

Find $\mathrm{fg}(4)$

The function $h$ is such that

$\mathrm{h} : x \rightarrow \text{from} \text{bar} 3 x^{2} - 12 x + 8$ where $x>2$

Express the inverse function $h^{-1}$ in the form $\mathrm{h}^{- 1} : x \rightarrow \text{from} bar....$

$\mathrm{h} ( x ) = \sqrt[3]{x}$ for all values of $x$

On the grid, draw the graph of the inverse function $y = \mathrm{h}^{- 1} ( x )$ for negative 2 less-than or slanted equal to x less-than or slanted equal to 2

Two functions, $f$ and $g$ are defined as

$\mathrm{f} : x \rightarrow \text{from} \text{bar} 1 + \frac{1}{x} \text{for} x > 0 \mathrm{g} : x \rightarrow \text{from} \text{bar} \frac{x + 1}{2} \text{for} x > 0$

Given that $h=\mathrm{fg}$ express the inverse function $h^{-1}$ in the form $\mathrm{h}^{- 1} : x \rightarrow \text{from} \text{bar} ...$

$\mathrm{h}^{- 1} : x \rightarrow \text{from} \text{bar}$.....................

For all values of $x$

$f(x)=\mathrm{sin}x\,g(x)=x+90$

On the grid, draw the graph of the composite function $y = \text{fg} ( x )$ for $0 ^{\circ} \text{less} - \text{than} or \text{slanted} \text{equal} to x \text{less} - \text{than} or \text{slanted} \text{equal} to 360 ^{\circ}$

$\mathrm{f} ( x ) = 2 x - 3$ and $\mathrm{g} ( x ) = x^{2}$

Show that $\mathrm{f}^{- 1} ( 55 ) = \mathrm{f} g ( 4 )$