Algebra
Linear functions
Zeros and roots of linear functions
20 practice questions
2 video lessons
Theory + worked examples
Theory
The zero (or root) of a function is an input where the output is \(0\):
\[f(x)=0.\]
For a line this is the \(x\)-intercept, found by solving \(mx+b=0\).
Zero, root, and \(x\)-intercept all name the same point.
The zero is the \(x\)-intercept.
Finding a zero.
Solving for the zero:
\[mx+b=0\ \Rightarrow\ x=-\dfrac{b}{m}\]
A non-vertical line has exactly one zero (unless horizontal).
How to find a zero
- Set \(f(x)=0\).
- Solve for \(x\).
- Write it as the point \((x,0)\).
- Confirm it is the \(x\)-intercept on the graph.
Example 1 — Find the zero
Find the zero of \(f(x)=2x-6\).
Solution
Set \(f(x)=0\).
| \(2x-6\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(3\) |
Example 2 — Zero as an intercept
What point on the graph is the zero \(x=3\)?
Solution
It is the \(x\)-intercept.
| \(\text{point}\) | \(=\) | \((3,0)\) |
Example 3 — Another zero
Find the root of \(f(x)=-x+4\).
Solution
Set equal to zero.
| \(-x+4\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(4\) |
Example 4 — Zero vs y-intercept
Is the zero the same as the \(y\)-intercept?
Solution
No — the zero is the \(x\)-intercept (where \(y=0\)); the \(y\)-intercept is where \(x=0\).
Common pitfalls
A zero is the \(x\)-intercept, not the \(y\)-intercept.
Set the function equal to \(0\), then solve.
The zero is an \(x\)-value (a point \((x,0)\)).
Frequently asked questions
What is a zero of a function?
An input where the output is \(0\).
Is a zero the same as an x-intercept?
Yes — they are the same point.
How do you find the zero of \(2x-6\)?
Solve \(2x-6=0\): \(x=3\).
Is the zero the y-intercept?
No — the zero is the \(x\)-intercept.
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