Functions and their graphs Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Functions and their graphs questions. See exactly how to solve problems on functions, composite, inverse, modulus.

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A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
The functions are defined by f(x)=2x+1f(x)=2 x + 1 and g(x)=x−3g(x)=x - 3. Find fg(5)fg(5), where fg(x)=f(g(x))fg(x)=f(g(x)).

Worked solution

  1. State the two functions

    f(x)=2x+1,g(x)=x−3f(x)=2 x + 1,\quad g(x)=x - 3

    We begin from the given definitions.

  2. Work out the inner function g(5)

    g(5)=2g(5)=2

    fg means f(g(x)), so evaluate the inner function first.

  3. State the value of fg(5)

    fg(5)=5fg(5)=5

    This is the required value.

Answer
55
Question 2
2 markseasy
The functions are defined by f(x)=x2f(x)=x^{2} and g(x)=x+1g(x)=x + 1. Find fg(2)fg(2), where fg(x)=f(g(x))fg(x)=f(g(x)).

Worked solution

  1. State the two functions

    f(x)=x2,g(x)=x+1f(x)=x^{2},\quad g(x)=x + 1

    We begin from the given definitions.

  2. Work out the inner function g(2)

    g(2)=3g(2)=3

    fg means f(g(x)), so evaluate the inner function first.

  3. State the value of fg(2)

    fg(2)=9fg(2)=9

    This is the required value.

Answer
99
Question 3
2 markseasy
The functions are defined by f(x)=3x−2f(x)=3 x - 2 and g(x)=x2g(x)=x^{2}. Find gf(2)gf(2), where gf(x)=g(f(x))gf(x)=g(f(x)).

Worked solution

  1. State the two functions

    f(x)=3x−2,g(x)=x2f(x)=3 x - 2,\quad g(x)=x^{2}

    We begin from the given definitions.

  2. Work out the inner function f(2)

    f(2)=4f(2)=4

    gf means g(f(x)), so evaluate the inner function first.

  3. State the value of gf(2)

    gf(2)=16gf(2)=16

    This is the required value.

Answer
1616
Question 4
2 markseasy
The functions are defined by f(x)=x+4f(x)=x + 4 and g(x)=2xg(x)=2 x. Find fg(3)fg(3), where fg(x)=f(g(x))fg(x)=f(g(x)).

Worked solution

  1. State the two functions

    f(x)=x+4,g(x)=2xf(x)=x + 4,\quad g(x)=2 x

    We begin from the given definitions.

  2. Work out the inner function g(3)

    g(3)=6g(3)=6

    fg means f(g(x)), so evaluate the inner function first.

  3. State the value of fg(3)

    fg(3)=10fg(3)=10

    This is the required value.

Answer
1010
Question 5
2 markseasy
The functions are defined by f(x)=5−xf(x)=5 - x and g(x)=x2g(x)=x^{2}. Find gf(2)gf(2), where gf(x)=g(f(x))gf(x)=g(f(x)).

Worked solution

  1. State the two functions

    f(x)=5−x,g(x)=x2f(x)=5 - x,\quad g(x)=x^{2}

    We begin from the given definitions.

  2. Work out the inner function f(2)

    f(2)=3f(2)=3

    gf means g(f(x)), so evaluate the inner function first.

  3. State the value of gf(2)

    gf(2)=9gf(2)=9

    This is the required value.

Answer
99

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