Hard A-Level Functions and their graphs Questions

Challenging, exam-style A-Level Functions and their graphs questions with worked solutions. Stretch yourself on the hardest functions, composite, solve, inverse problems.

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A-Level34 questionsStep-by-step solutions
Question 1
8 markschallenging
State the range of f(x)=1x2+1f(x)=\frac{1}{x^{2} + 1}.
Show worked solution

Worked solution

  1. State the function

    f(x)=1x2+1f(x)=\frac{1}{x^{2} + 1}

    We begin from the given definition.

  2. Determine the possible output values

    x2+1≥1 so 0<1x2+1≤1x^2+1\ge 1 \text{ so } 0<\tfrac{1}{x^2+1}\le 1

    The range is the set of values the function can take.

  3. Locate the turning value

    maximum at x=0\text{maximum at } x=0

    The extreme value bounds the range.

  4. Consider the behaviour for large |x|

    examine the outputs as x→±∞\text{examine the outputs as } x \to \pm\infty

    The end behaviour helps fix the extent of the range.

  5. Combine the bound with the reachable values

    x2+1≥1 so 0<1x2+1≤1x^2+1\ge 1 \text{ so } 0<\tfrac{1}{x^2+1}\le 1

    The extreme value together with the trend fixes the range.

  6. Substitute x=-3 as a numerical check

    x=−3: 110x=-3:\ \frac{1}{10}

    Evaluating at a point provides a check on the result.

  7. Substitute x=-2 as a numerical check

    x=−2: 15x=-2:\ \frac{1}{5}

    Evaluating at a point provides a check on the result.

  8. Substitute x=-1 as a numerical check

    x=−1: 12x=-1:\ \frac{1}{2}

    Evaluating at a point provides a check on the result.

  9. Substitute x=0 as a numerical check

    x=0: 1x=0:\ 1

    Evaluating at a point provides a check on the result.

  10. Substitute x=1 as a numerical check

    x=1: 12x=1:\ \frac{1}{2}

    Evaluating at a point provides a check on the result.

  11. Substitute x=2 as a numerical check

    x=2: 15x=2:\ \frac{1}{5}

    Evaluating at a point provides a check on the result.

  12. Substitute x=3 as a numerical check

    x=3: 110x=3:\ \frac{1}{10}

    Evaluating at a point provides a check on the result.

  13. Substitute x=4 as a numerical check

    x=4: 117x=4:\ \frac{1}{17}

    Evaluating at a point provides a check on the result.

  14. Substitute x=5 as a numerical check

    x=5: 126x=5:\ \frac{1}{26}

    Evaluating at a point provides a check on the result.

  15. State the answer

    0<f(x)≤10 < f(x) \le 1

    This is the range of the function.

Answer
0<f(x)≤10 < f(x) \le 1
Question 2
8 markschallenging
State the range of f(x)=2−∣x+1∣f(x)=2 - \left|{x + 1}\right|.
Show worked solution

Worked solution

  1. State the function

    f(x)=2−∣x+1∣f(x)=2 - \left|{x + 1}\right|

    We begin from the given definition.

  2. Determine the possible output values

    ∣x+1∣≥0 so 2−∣x+1∣≤2\left|x+1\right|\ge 0 \text{ so } 2-\left|x+1\right|\le 2

    The range is the set of values the function can take.

  3. Locate the turning value

    maximum at x=−1\text{maximum at } x=-1

    The extreme value bounds the range.

  4. Consider the behaviour for large |x|

    examine the outputs as x→±∞\text{examine the outputs as } x \to \pm\infty

    The end behaviour helps fix the extent of the range.

  5. Combine the bound with the reachable values

    ∣x+1∣≥0 so 2−∣x+1∣≤2\left|x+1\right|\ge 0 \text{ so } 2-\left|x+1\right|\le 2

    The extreme value together with the trend fixes the range.

  6. Substitute x=-3 as a numerical check

    x=−3: 0x=-3:\ 0

    Evaluating at a point provides a check on the result.

  7. Substitute x=-2 as a numerical check

    x=−2: 1x=-2:\ 1

    Evaluating at a point provides a check on the result.

  8. Substitute x=-1 as a numerical check

    x=−1: 2x=-1:\ 2

    Evaluating at a point provides a check on the result.

  9. Substitute x=0 as a numerical check

    x=0: 1x=0:\ 1

    Evaluating at a point provides a check on the result.

  10. Substitute x=1 as a numerical check

    x=1: 0x=1:\ 0

    Evaluating at a point provides a check on the result.

  11. Substitute x=2 as a numerical check

    x=2: −1x=2:\ -1

    Evaluating at a point provides a check on the result.

  12. Substitute x=3 as a numerical check

    x=3: −2x=3:\ -2

    Evaluating at a point provides a check on the result.

  13. Substitute x=4 as a numerical check

    x=4: −3x=4:\ -3

    Evaluating at a point provides a check on the result.

  14. Substitute x=5 as a numerical check

    x=5: −4x=5:\ -4

    Evaluating at a point provides a check on the result.

  15. State the answer

    f(x)≤2f(x) \le 2

    This is the range of the function.

Answer
f(x)≤2f(x) \le 2
Question 3
8 markschallenging
State the domain of f(x)=9−x2f(x)=\sqrt{9 - x^{2}}.
Show worked solution

Worked solution

  1. State the function

    f(x)=9−x2f(x)=\sqrt{9 - x^{2}}

    We begin from the given definition.

  2. Identify where the function is defined

    9−x2≥09-x^2\ge 0

    Look for restrictions such as square roots, logarithms or division by zero.

  3. Consider the excluded values

    values making the expression undefined are removed\text{values making the expression undefined are removed}

    Any input causing an undefined output is excluded from the domain.

  4. Express the restriction as an inequality

    9−x2≥09-x^2\ge 0

    Rearranging the condition gives the set of valid inputs.

  5. Substitute x=-3 as a numerical check

    x=−3: 0x=-3:\ 0

    Evaluating at a point provides a check on the result.

  6. Substitute x=-2 as a numerical check

    x=−2: 5x=-2:\ \sqrt{5}

    Evaluating at a point provides a check on the result.

  7. Substitute x=-1 as a numerical check

    x=−1: 22x=-1:\ 2 \sqrt{2}

    Evaluating at a point provides a check on the result.

  8. Substitute x=0 as a numerical check

    x=0: 3x=0:\ 3

    Evaluating at a point provides a check on the result.

  9. Substitute x=1 as a numerical check

    x=1: 22x=1:\ 2 \sqrt{2}

    Evaluating at a point provides a check on the result.

  10. Substitute x=2 as a numerical check

    x=2: 5x=2:\ \sqrt{5}

    Evaluating at a point provides a check on the result.

  11. Substitute x=3 as a numerical check

    x=3: 0x=3:\ 0

    Evaluating at a point provides a check on the result.

  12. Recall the meaning of the domain

    domain={x:f(x) is defined}\text{domain} = \{x : f(x) \text{ is defined}\}

    The domain is the set of permitted inputs.

  13. Recall the meaning of the range

    range={f(x):x∈domain}\text{range} = \{f(x) : x \in \text{domain}\}

    The range is the set of achievable outputs.

  14. Recall the composite-function convention

    fg(x)=f(g(x))fg(x)=f\left(g(x)\right)

    The inner function is applied first.

  15. State the answer

    −3≤x≤3-3 \le x \le 3

    This is the domain of the function.

Answer
−3≤x≤3-3 \le x \le 3
Question 4
8 markschallenging
Given f(x)=x+2xf(x)=\frac{x + 2}{x} and g(x)=x−1g(x)=x - 1, which of the following is fg(x)fg(x)?
Show worked solution

Worked solution

  1. State the two functions

    f(x)=x+2x,g(x)=x−1f(x)=\frac{x + 2}{x},\quad g(x)=x - 1

    We begin from the given definitions.

  2. Form fg(x)=f(g(x))

    fg(x)=f(x−1)fg(x)=f\left(x - 1\right)

    Substitute g into f, not the other way round.

  3. Carry out the substitution

    fg(x)=x+1x−1fg(x)=\frac{x + 1}{x - 1}

    Replace x in f by the whole of g.

  4. Contrast with gf(x)

    gf(x)=2xgf(x)=\frac{2}{x}

    gf is generally different from fg, a common trap.

  5. Substitute x=-3 as a numerical check

    x=−3: 12x=-3:\ \frac{1}{2}

    Evaluating at a point provides a check on the result.

  6. Substitute x=-2 as a numerical check

    x=−2: 13x=-2:\ \frac{1}{3}

    Evaluating at a point provides a check on the result.

  7. Substitute x=-1 as a numerical check

    x=−1: 0x=-1:\ 0

    Evaluating at a point provides a check on the result.

  8. Substitute x=0 as a numerical check

    x=0: −1x=0:\ -1

    Evaluating at a point provides a check on the result.

  9. Substitute x=2 as a numerical check

    x=2: 3x=2:\ 3

    Evaluating at a point provides a check on the result.

  10. Substitute x=3 as a numerical check

    x=3: 2x=3:\ 2

    Evaluating at a point provides a check on the result.

  11. Substitute x=4 as a numerical check

    x=4: 53x=4:\ \frac{5}{3}

    Evaluating at a point provides a check on the result.

  12. Substitute x=5 as a numerical check

    x=5: 32x=5:\ \frac{3}{2}

    Evaluating at a point provides a check on the result.

  13. Substitute x=6 as a numerical check

    x=6: 75x=6:\ \frac{7}{5}

    Evaluating at a point provides a check on the result.

  14. Substitute x=7 as a numerical check

    x=7: 43x=7:\ \frac{4}{3}

    Evaluating at a point provides a check on the result.

  15. Select the correct composite

    fg(x)=x+1x−1fg(x)=\frac{x + 1}{x - 1}

    This is f(g(x)) in simplest form.

Answer
fg(x)=x+1x−1fg(x)=\frac{x + 1}{x - 1}
Question 5
8 markschallenging
Given f(x)=x2−xf(x)=x^{2} - x and g(x)=2x+1g(x)=2 x + 1, which of the following is fg(x)fg(x)?
Show worked solution

Worked solution

  1. State the two functions

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

    We begin from the given definitions.

  2. Form fg(x)=f(g(x))

    fg(x)=f(2x+1)fg(x)=f\left(2 x + 1\right)

    Substitute g into f, not the other way round.

  3. Carry out the substitution

    fg(x)=−2x+(2x+1)2−1fg(x)=- 2 x + \left(2 x + 1\right)^{2} - 1

    Replace x in f by the whole of g.

  4. Simplify

    fg(x)=2x(2x+1)fg(x)=2 x \left(2 x + 1\right)

    Collecting terms gives the composite.

  5. Contrast with gf(x)

    gf(x)=2x2−2x+1gf(x)=2 x^{2} - 2 x + 1

    gf is generally different from fg, a common trap.

  6. Substitute x=-3 as a numerical check

    x=−3: 30x=-3:\ 30

    Evaluating at a point provides a check on the result.

  7. Substitute x=-2 as a numerical check

    x=−2: 12x=-2:\ 12

    Evaluating at a point provides a check on the result.

  8. Substitute x=-1 as a numerical check

    x=−1: 2x=-1:\ 2

    Evaluating at a point provides a check on the result.

  9. Substitute x=0 as a numerical check

    x=0: 0x=0:\ 0

    Evaluating at a point provides a check on the result.

  10. Substitute x=1 as a numerical check

    x=1: 6x=1:\ 6

    Evaluating at a point provides a check on the result.

  11. Substitute x=2 as a numerical check

    x=2: 20x=2:\ 20

    Evaluating at a point provides a check on the result.

  12. Substitute x=3 as a numerical check

    x=3: 42x=3:\ 42

    Evaluating at a point provides a check on the result.

  13. Substitute x=4 as a numerical check

    x=4: 72x=4:\ 72

    Evaluating at a point provides a check on the result.

  14. Substitute x=5 as a numerical check

    x=5: 110x=5:\ 110

    Evaluating at a point provides a check on the result.

  15. Select the correct composite

    fg(x)=2x(2x+1)fg(x)=2 x \left(2 x + 1\right)

    This is f(g(x)) in simplest form.

Answer
fg(x)=2x(2x+1)fg(x)=2 x \left(2 x + 1\right)

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