Function Notation (new GCSE Maths) - YouTube - Free Printable

Let me work through each part of this problem step by step.

Given:
- f(x) = 5x + 3
- g(x) = 3(x + 1)

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Part a) Find f(4)

To find f(4), I substitute x = 4 into the function f(x):
f(4) = 5(4) + 3
f(4) = 20 + 3
f(4) = 23

✓ This matches what's shown in the image.

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Part b) Solve f(x) = 23

I need to solve: 5x + 3 = 23

Step 1: Subtract 3 from both sides
5x = 23 - 3
5x = 20

Step 2: Divide both sides by 5
x = 20 ÷ 5
x = 4

✓ This also matches what's shown in the image.

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Part c) Find f⁻¹(x) (the inverse function)

To find the inverse, I start with y = f(x) and solve for x in terms of y.

Step 1: Start with y = 5x + 3

Step 2: Subtract 3 from both sides
y - 3 = 5x

Step 3: Divide both sides by 5
(y - 3)/5 = x

Step 4: Swap x and y (to write it as a function of x)
y = (x - 3)/5

So f¹(x) = (x - 3)/5

✓ This matches what's shown in the image.

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Part d) State fg(x)

This means f(g(x)) - we need to compose the functions.

Step 1: First, let me recall that g(x) = 3(x + 1) = 3x + 3

Step 2: Now substitute g(x) into f(x)
f(g(x)) = f(3x + 3)

Step 3: Apply the function f to (3x + 3)
f(3x + 3) = 5(3x + 3) + 3

Step 4: Expand and simplify
= 15x + 15 + 3
= 15x + 18

Let me verify this calculation:
- g(x) = 3(x + 1) = 3x + 3 ✓
- f(g(x)) = 5(3x + 3) + 3
- = 5 × 3x + 5 × 3 + 3
- = 15x + 15 + 3
- = 15x + 18 ✓

Final Answer:
a) f(4) = 23
b) x = 4
c) f⁻¹(x) = (x - 3)/5
d) fg(x) = 15x + 18