GCSE(H),

An inverse function reverses the effects of the original function. It is written as f^{-1}(x).

If f(*x*) = 3*x* + 2, determine the inverse function for f(*x*):

The original function is:

→x |
x3 | +2 | → 3x+2 |

Reverse the operations, starting with *x*:

`frac((x-2))(3)` | ÷3 | -2 | x ← |

Therefore f^{-1}(x) = `frac(x-2)(3)`. If a function is performed on a value; and the inverse function is applied to the result, then the original value is obtained. This is a useful check to ensure that the inverse function has been correctly determined. Check with a random number: use f(3) to obtain a result of 11. Then use the result in the inverse function, so f^{-1}(11) = `frac((11-2))(3)` = 3.

1. Find the inverse function of f(*x*) = 5(x - 1)

Answer: f^{-1}(x) = `frac(x)(5)` + 1

Function: *x* → - 1 → x 5 → 5(*x* -1)

Inverse: `frac(x)(5)` + 1 ← + 1 ← ÷ 5 ← *x*

Check: use 3 in the original function: 5(3 - 1) = 10. Use 10 in the inverse function: 10 ÷ 5 + 1 = 3.

2. If f(*x*) = *x*^{2} + 5, what is the inverse function?

Answer: √(*x* - 5)

Function: *x* → *x*^{2} → + 5 →*x*^{2} + 5

Inverse: √(*x* - 5) ← √ ← - 5 ← *x*

Check: use 3 in the original function: 3^{2} + 5 = 14. Try f^{-1}(14): √(14 - 5) = √9 = 3.

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