Many Functions questions give you a rule with something missing, then ask you to find it. The method is to turn what the question says into an equation, then solve and check.
This is the last lesson in the Functions chapter. It uses composite functions and inverse functions.
What kinds of unknown can appear?
| Type | The question gives | You set up |
|---|---|---|
| Unknown constant | f(2) = 7 and f(x) = ax + 3 | 2a + 3 = 7 |
| Composite equals a value | fg(x) = 11 | the composite, equal to 11 |
| Function equals its inverse | f(x) = f⁻¹(x) | two expressions, equal |
Worked example 1: an unknown constant in a composite
Given f(x) = 4x − 1 and g(x) = x + a, and fg(2) = 15. Find a.
- fg(2) = f(g(2)). First g(2) = 2 + a.
- Then f(2 + a) = 4(2 + a) − 1.
- Set 4(2 + a) − 1 = 15, so 4(2 + a) = 16 and 2 + a = 4.
- So a = 2. Check: g(2) = 4 and f(4) = 15.
Worked example 2: f equals its own inverse
Given f(x) = (x + 4) ÷ 3. Solve f(x) = f⁻¹(x).
First find the inverse: y = (x + 4) ÷ 3 gives x = 3y − 4, so f⁻¹(x) = 3x − 4.
Set (x + 4) ÷ 3 = 3x − 4. Multiply by 3: x + 4 = 9x − 12, so 16 = 8x and x = 2.
Check: f(2) = 6 ÷ 3 = 2 and f⁻¹(2) = 6 − 4 = 2. Both equal 2, so the answer stands.
Worked example 3: rejecting a value
Given f(x) = x + 2 and g(x) = x², with domain x > 0. Solve gf(x) = 25.
gf(x) = (x + 2)². Set (x + 2)² = 25, so x + 2 = 5 or x + 2 = −5. This gives x = 3 or x = −7.
The domain is x > 0, so x = −7 is rejected. The answer is x = 3. Check: f(3) = 5 and g(5) = 25.
The mistake that costs marks
The slip is to solve fg(x) when the question says gf(x), or to keep a root the domain forbids. Both come from not underlining what the question asks before starting. Write the composite you need on the first line, for example “gf(x) = g(f(x))”.
Check yourself
Given f(x) = x − 1 and g(x) = 2x². Solve gf(x) = 18.
Answer
gf(x) = g(f(x)) = 2(x − 1)². Set 2(x − 1)² = 18, so (x − 1)² = 9 and x − 1 = 3 or −3.
So x = 4 or x = −2, because no domain restriction is given.
Check x = 4: f(4) = 3 and g(3) = 18. Check x = −2: f(−2) = −3 and g(−3) = 18.
What to study next
Test the whole chapter with the functions practice set. If you are unsure how to begin an unfamiliar question, read choosing the right method in an unfamiliar Add Maths question.
If you want a teacher to work through these question types with you, see online one-to-one Additional Mathematics tuition.