Introduction to Functions and Graphs
A function takes each input and turns it into exactly one output. Put x = 3 into the rule f(x) = 2x + 1 to get f(3) = 7. Show it with a table of values, then plot the points as a graph.
Video Lesson
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Flashcards
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What Does Function Notation f(x) Mean?
- shows the output when the input is x.
- To find , substitute into the rule.
- For example, if , then .
How to Graph a Function?
- First create a table by choosing x-values and finding the matching f(x) values.
- Then plot the points on a graph and join them with a straight or smooth line.
Does a Graph Represent a Function?
- A graph is a function if each x-value has only one y-value.
- If one x-value has more than one y-value, it is not a function.
Reading Function Values from Graphs
- To find f(x), go to the x-value and read the y-value on the graph.
- Solving means the y-value is zero. This happens where the graph crosses the x-axis.
Practice Questions
Test your understanding
What is the output of the function when ?
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Substituting means replacing x in the rule with the value given.
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The multiplication is done before the addition, so work out first.
What is the output of the function when ?
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Replace x in the rule with 2.
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Multiply first, then subtract; the minus sign belongs to the 3.
For the function , what is the value of ?
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The 3 inside the brackets means we replace x with 3.
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Do the multiplication first, then take away 4.
For the function , what is the value of x when ?
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Substitute 7 for to get the equation .
Add 5 to both sides: .
Divide both sides by 3: .
Check: , as required.
For the function , what is the value of ?
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Substitute into the rule, keeping the negative in brackets:
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Multiplying 5 by a negative number gives a negative answer, and adding the 8 then brings it back up to . Stopping at misses the last step.
Based on the graph, what are the values of when ?
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Draw the horizontal line across the graph and look for every place it meets the curve.
It touches the curve at the turning point where , the curve drops steeply through it at , and the curve rises back through it at .
A touching point counts as a solution just like a crossing, so has three solutions: , and .
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Interactive Activity
Introduction to functions and graphs
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Frequently asked questions
The questions students bump into most on this topic
A function is like a machine: you put a number in, it applies a rule, and one number comes out. The rule is that each input gives exactly one output. f(x) = 2x + 1 is a function: an input of 1 gives 3, an input of 2 gives 5. Its graph is the picture of all the input-output pairs.
f stands for the function and x for the input; whatever comes after the equals sign is the output. For f(x) = 2x + 1, f(3) means “the output when the input is 3”: 2 × 3 + 1 = 7, so f(3) = 7. The notation lets you name an input and its output in one short line, which is why exams use it.
Three kinds of question. Evaluate: substitute the input, so for f(x) = 3x + 2, f(5) = 3 × 5 + 2 = 17. Solve: set the rule equal to the output and find x, so f(x) = 3x − 5 = 7 gives 3x = 12 and x = 4. Read a graph: find the x-value, go up to the curve and read off y. Every GCSE functions question is one of these.
Check whether any x-value has more than one y-value. If a single input lines up with two or more points on the graph, it is not a function, because a function gives exactly one output for each input. A quick way to see it: if a vertical line could cross the graph more than once, it is not a function.
y is the output of f(x), so read y off the graph. To find f(−2), find x = −2 on the horizontal axis, go to the curve and read the y-value: in the video it is −3, so f(−2) = −3. To solve f(x) = 0, look where the graph crosses the x-axis, because that is where y is 0: at x = −4, 0 and 5.
Yes, and it is still a function. The rule is one output for each input, not one input for each output. In the video’s graph, f(x) = 0 at three different inputs, x = −4, 0 and 5, and that is perfectly fine. What is not allowed is one input with two outputs.
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