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      Checking and securing understanding of function notation

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      Lesson details

      Learning outcome

      I can define and use function notation.

      Key learning points

      1. There is a shorthand for writing a function
      2. The notation means that the whole function does not need to be written out each time
      3. The notation should always be defined at the start
      4. It is possible to substitute the notation for the function it represents

      Keywords

      • Function - A function is a mathematical relationship that uniquely maps values of one set to the values of another set.

      • Domain - The domain of a function is the set of values that the mapping is performed on.

      • Range - The range of a function is the set of values mapped to by the function and the stated domain.

      • Inverse function - An inverse function reverses the mapping of the original function.

      Common misconception

      Pupils may confuse the domain and the range.

      Drawing the graph of the function and labelling the axes correctly will help pupils connect domain and range.

      Teacher tip

      If pupils are struggling to know whether a relationship is a valid function, encourage them to draw the graph as the visual representation can help.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
       except where otherwise stated. See Oak's terms & conditions
       (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      Download starter quiz questions (PDF)

      6 Questions

      Q1.
      When graphing the linear equation $$y=6x-10$$, what is the value of the $$y$$ coordinate when $$x=3$$?

      Correct Answer: 8, y=8

      Q2.
      When graphing the linear equation $$y=6x-10$$, what is the value of the $$y$$ coordinate when $$x=-1$$?

      Correct Answer: -16

      Q3.
      Match these key words to their definitions.

      Correct Answer:domain,the set of values for a function that the mapping is performed on

      the set of values for a function that the mapping is performed on

      Correct Answer:function,a relationship which uniquely maps values in one set to another

      a relationship which uniquely maps values in one set to another

      Correct Answer:inverse function ,a relationship which reverses the mapping of the original function

      a relationship which reverses the mapping of the original function

      Correct Answer:range,the set of values mapped to, by the function

      the set of values mapped to, by the function

      Q4.
      When we graph the equation $$y= x^2$$, what is the minimum value (i.e. the smallest value of the $$y$$ coordinate)?

      Correct Answer: 0, y=0

      Q5.
      Which of these is not a valid input for $$x$$ in the equation $$y = \frac{1}{2x}$$ ?

      -2
      -1
      Correct answer: 0
      1
      2

      Q6.
      What does $$\text{sin}^{-1} (x)$$ mean?

      "sine of -1"
      "sine multiplied by -1"
      "1 divided by $$\text{sin}(x)$$"
      Correct answer: "sine inverse of $$x$$"
      "$$\text{sin}(x)$$ to the power of -1"

      Assessment exit quiz

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      6 Questions

      Q1.
      If $$\text{f}(x)= 4(x-2)$$, which is the correct notation for the inverse function $$\text{f}$$ inverse of $$x$$ ?

      $$\text{f}(-1)= \frac{x}{4} + 2$$
      $$-\text{f}(x)= \frac{x}{4} + 2$$
      Correct answer: $$\text{f}^{-1}(x)= \frac{x}{4} + 2$$
      $$\text{f}(x)^{-1}= \frac{x}{4} + 2$$

      Q2.
      If $$\text{g}(4) = 6$$, then $$\text{g}^{-1}(6) =$$ .

      Correct Answer: 4

      Q3.
      If $$\text{f}(x)= 3x - 5$$, $$\text{g}(x)= 10-2x$$ and $$\text{h}(x)= x^2 + 3$$. Match up these functions with their outputs.

      Correct Answer:$$\text{f}(-2)$$,$$-11$$

      $$-11$$

      Correct Answer:$$\text{g}(-2)$$,$$14$$

      $$14$$

      Correct Answer:$$\text{h}(-2)$$,$$7$$

      $$7$$

      Correct Answer:$$\text{f}(1)$$,$$-2$$

      $$-2$$

      Correct Answer:$$\text{g}(1)$$,$$8$$

      $$8$$

      Correct Answer:$$\text{h}(1)$$,$$4$$

      $$4$$

      Q4.
      Which is the correct range for $$\text{f}(x)= x^2 + 4$$ with the domain {$$x$$ : $$x$$ is any value}?

      {$$\text{f}(x) : \text{f}(x) > 4$$}
      Correct answer: {$$\text{f}(x) : \text{f}(x) \ge 4$$}
      {$$\text{f}(x) : \text{f}(x) < 4$$}
      {$$\text{f}(x) : \text{f}(x) \le 4$$}

      Q5.
      Which is the correct range for $$\text{f}(x)= \text{sin}(x)$$ with the domain {$$x : 0 \le x \le 180$$}?

      {$$\text{f}(x) : 0 \le \text{f}(x) \le 0$$}
      Correct answer: {$$\text{f}(x) : 0 \le \text{f}(x) \le 1$$}
      {$$\text{f}(x) : -1 \le \text{f}(x) \le 1$$}
      {$$\text{f}(x) : -1 \le \text{f}(x)\le 0$$}

      Q6.
      Which domain restrictions would make $$\text{f}(x)= \frac{24}{x}$$ a valid function?

      Correct answer: $$\text{f}(x)= \frac{24}{x}$$ when {$$x : x > 0$$}
      $$\text{f}(x)= \frac{24}{x}$$ when {$$x : x \ge 0$$}
      Correct answer: $$\text{f}(x)= \frac{24}{x}$$ when {$$x : x < 0$$}
      $$\text{f}(x)= \frac{24}{x}$$ when {$$x : x \le 0$$}
      Correct answer: $$\text{f}(x)= \frac{24}{x}$$ when {$$x : x \not= 0$$}

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