Understand functions as input → output rules · lesson

Following a rule

💡 A function rule tells you what to do to the input to get the output.

Think of a function as a machine. You put in an input. The rule acts on it. Out comes the output.

Follow the steps in the right order. Use the order of operations.

🧠 Let's work it out

A rule triples the input and then subtracts 1. What is the output for 6?

What do you think comes next?

💡 It's a function only if each input always gives the same single output.

For a function, each input has exactly one output. If the same input ever gives two different outputs, it is not a function.

Two different inputs can share an output. That is fine.

In a list of pairs (x, y), look for a repeated x with different y's.

🧠 Let's work it out

Is {(2, 4), (3, 4), (5, 1)} a function?

What do you think comes next?

💡 If any up-and-down line hits a graph more than once, the graph is not a function.

On a graph, each x is an input. If one x has two points above or below it, that input has two outputs.

So slide a vertical line across the graph. If it ever hits two points at once, it is not a function. This is the vertical line test.

A graph that is itself a vertical line, like x = −2, fails the test. Every point on it has the same input, −2, so that one input has many outputs.

🧠 Let's work it out

A graph is a circle. Is it a function?

Draw a vertical line through the middle.
It hits the circle at the top and the bottom: two points with the same x.
So a circle is not a function.
Compare: a slanted straight line passes the test, since each vertical line hits it once.

What do you think comes next?

vertical line test: a graph is a function if no up-and-down line hits it twice

💡 For a linear table, find how much the output changes per 1 input, then extend the pattern.

Look at how the output changes when the input goes up by 1. If it changes by the same amount each time, the rule is linear.

The rule is output = (change per 1) × input + (output at 0).

If the inputs jump by 2, divide the change by 2 to get the change per 1.

🧠 Let's work it out

A table shows 1 → 4, 2 → 7, 3 → 10. What is the output for input 8?

What do you think comes next?

💡 To find the input for a given output, set the rule equal to the output and solve.

Sometimes you know the output and need the input. Write the rule as an equation with the output filled in.

Then undo the steps to solve for the input.

🧠 Let's work it out

The rule is output = 4 × input + 3. What input gives 31?

What do you think comes next?