The functions f and g are such that f(x) = 5x + 3 g(x) = ax + b where a and b are constants - Edexcel - GCSE Maths - Question 22 - 2017 - Paper 2

Next, we need to find the inverse of f(x). Starting with:

$y = 5x + 3$

We can solve for x:

1. Subtract 3 from both sides: $y - 3 = 5x$

2. Divide by 5: $x = \frac{y - 3}{5}$

Thus, the inverse function is:

$f^{-1}(x) = \frac{x - 3}{5}$

Now we need to find f^{-1}(33):

$f^{-1}(33) = \frac{33 - 3}{5} = \frac{30}{5} = 6$

Setting this equal to g(1):

$g(1) = a(1) + b = a + b \Rightarrow 6 = a + b$

This gives us a second equation.

We now have the system of equations:

1. $3a + b = 20$
2. $a + b = 6$

Subtracting the second equation from the first, we get:

$(3a + b) - (a + b) = 20 - 6$ $2a = 14 \Rightarrow a = 7$

Substituting the value of a back into the second equation:

$7 + b = 6 \Rightarrow b = 6 - 7 \Rightarrow b = -1$