Worked examples
The question
Question: find x when f(x) = 7 for f(x) = 2x − 3.
f(x) = 2x − 3
f(x) = 7 means: the answer that comes out is 7.
We look for the x that goes in.
f(x) = 7: 7 comes out
find what goes in
Careful!
It is not f(7).
7 is what comes out, not what goes in.
Careful: not f(7) ✗
1 Write the equation
f(x) is 2x − 3.
So 2x − 3 = 7.
Now we solve it.
2x − 3 = 7
We want x by itself.
Two things are stuck to x: times 2 and − 3.
The − 3 is on the outside, so it comes off first.
Then the times 2.
2 Remove the − 3
To remove − 3, we add 3.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
2x − 3 + 3 = 7 + 3
− 3 and + 3 make zero.
Cross them out.
7 + 3 = 10.
7 + 3 = 10
Now we have 2x = 10.
2x = 10
3 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2x2 = 102
2 divided by 2 is 1.
So just x is left.
10 divided by 2 is 5.
102 = 5
So x = 5!
x = 5
4 Check
Let's check: put 5 into f.
We get 7.
f(5) = 2(5) − 3 = 10 − 3 = 7
So f(5) = 7.
x = 5 is right!
Let's look back.
First: write the equation.
Then: remove the − 3.
Then: remove the 2.
Last: check.
▶ Watch it solved
The question
Question: find x when f(x) = 11 for f(x) = −2x + 5.
f(x) = −2x + 5
f(x) = 11 means: the answer that comes out is 11.
We look for the x that goes in.
f(x) = 11: 11 comes out
find what goes in
Careful!
It is not f(11).
11 is what comes out, not what goes in.
Careful: not f(11) ✗
1 Write the equation
f(x) is −2x + 5.
So −2x + 5 = 11.
Now we solve it.
−2x + 5 = 11
We want x by itself.
Two things are stuck to x: times −2 and + 5.
The + 5 is on the outside, so it comes off first.
Then the times −2.
2 Remove the + 5
To remove + 5, we take away 5.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
−2x + 5 − 5 = 11 − 5
+ 5 and − 5 make zero.
Cross them out.
11 − 5 = 6.
11 − 5 = 6
Now we have −2x = 6.
−2x = 6
3 Remove the −2
−2x means −2 times x.
We want just x.
To remove times −2, we divide by −2.
We do it on both sides, so they stay equal.
−2x−2 = 6−2
−2 divided by −2 is 1.
So just x is left.
6 divided by −2 is −3.
6−2 = −3
So x = −3!
x = −3
4 Check
Let's check: put −3 into f.
We get 11.
f(−3) = −2(−3) + 5 = 6 + 5 = 11
So f(−3) = 11.
x = −3 is right!
Let's look back.
First: write the equation.
Then: remove the + 5.
Then: remove the −2.
Last: check.
▶ Watch it solved
The question
Question: find x when f(x) = 25 for f(x) = 3x² − 2.
f(x) = 3x2 − 2
f(x) = 25 means: the answer that comes out is 25.
We look for the x that goes in.
f(x) = 25: 25 comes out
find what goes in
Careful!
It is not f(25).
25 is what comes out, not what goes in.
Careful: not f(25) ✗
1 Write the equation
f(x) is 3x² − 2.
So 3x² − 2 = 25.
Now we solve it.
3x2 − 2 = 25
2 Get x² alone
There is no x term, so we get x² by itself.
To remove the − 2, we add 2 on both sides.
3x2 − 2 + 2 = 25 + 2
25 + 2 = 27.
Now we have 3x² = 27.
3x2 = 27
To remove times 3, we divide both sides by 3.
3x23 = 273
27 divided by 3 is 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 25.
f(3) = 3(3)2 − 2 = 27 − 2 = 25
And −3 into f.
We get 25 too.
f(−3) = 3(−3)2 − 2 = 27 − 2 = 25
So f(3) = 25 and f(−3) = 25.
Both answers are right!
Let's look back.
First: write the equation.
Then: get x² alone.
Then: square root of both sides.
Last: check.
▶ Watch it solved
