Step by step
The question
Question: find x when f(x) = 8 for f(x) = x² − 1.
f(x) = x2 − 1
f(x) = 8 means: the answer that comes out is 8.
We look for the x that goes in.
f(x) = 8: 8 comes out
find what goes in
Careful!
It is not f(8).
8 is what comes out, not what goes in.
Careful: not f(8) ✗
1 Write the equation
f(x) is x² − 1.
So x² − 1 = 8.
Now we solve it.
x2 − 1 = 8
2 Get x² alone
There is no x term, so we get x² by itself.
To remove the − 1, we add 1 on both sides.
x2 − 1 + 1 = 8 + 1
8 + 1 = 9.
Now we have x² = 9.
x2 = 9
3 Square root of both sides
x squared is 9.
So x is the square root of 9, or minus it.
Minus times minus is plus, so the negative one works too.
x² = k → x = √k or x = −√k
3 × 3 = 9, (−3) × (−3) = 9
3 times 3 is 9.
So x = 3 or x = −3.
x = 3 or x = −3
4 Check
Let's check: put 3 into f.
We get 8.
f(3) = 32 − 1 = 9 − 1 = 8
And −3 into f.
We get 8 too.
f(−3) = (−3)2 − 1 = 9 − 1 = 8
So f(3) = 8 and f(−3) = 8.
Both answers are right!
Let's look back.
First: write the equation.
Then: get x² alone.
Then: square root of both sides.
Last: check.
Answer
x = 3 or x = −3