Critical help please. Level 3 maths algebraic methods is really hard.

Q)
Two functions f and g are defined by
f:x (arrow) x^2 + 3
g:x (arrow) 2x+1

Find
a) the function fg
b) solve the equation f(x)=12g^-1(x)

I need to show my working out, please help someone

by fg I assume you mean f*g

f*g = (x^2+3)*(2x+1) = 2x^3 + x^2 + 6x + 3

g=2x+1, so g^-1 = (x-1)/2
so, if f = 12 g^-1,
x^2+3 = 12 * (x-1)/2
x^2+3 = 6x-6
x^2 - 6x + 9 = 0
(x-3)^2 = 0
x = 3

Thank you steve

It says i need to find f(g), i got a different result for some reason help please!

And also in the second part, how did you get g^-1 = (x-1)/2

Thank you steve x

That's because fg is ambiguous. I did f(x) * g(x).

f◦g = f(g) = g^2+3 = (2x+1)^2+3
= 4x^2 + 4x + 4

to get g^-1, just solve for x

g = 2x+1
g-1 = 2x
x = (g-1)/2
then switch variables, and g^-1 = (x-1)/2

Is there any reason why f(g)=g^2+3

why do we replace the x with g?

Sorry steve im trying my hardest to understand this

Sure! I can help you with your math problem. Let's go step by step.

a) To find the composite function fg, you need to substitute g(x) into f(x). In other words, replace every occurrence of x in f(x) with the expression for g(x).

Step 1: Write the function f(x) = x^2 + 3.
Step 2: Replace x in f(x) with g(x) = 2x + 1.
So, f(g(x)) = (2x + 1)^2 + 3.

To simplify this further, you can expand the equation:
f(g(x)) = (2x + 1)(2x + 1) + 3.

Multiply using the FOIL method:
f(g(x)) = 4x^2 + 2x + 2x + 1 + 3.
f(g(x)) = 4x^2 + 4x + 4.

Therefore, the function fg is f(g(x)) = 4x^2 + 4x + 4.

b) To solve the equation f(x) = 12g^-1(x), you need to find the inverse of g(x) and substitute it into f(x) to equate it with 12.

Step 1: Write the function g(x) = 2x + 1.
Step 2: To find the inverse of g(x), interchange x and y and solve for y.
So, x = 2y + 1.
Rearranging the equation, we get y = (x - 1) / 2.

The inverse of g(x) is g^(-1)(x) = (x - 1) / 2.

Step 3: Substitute g^(-1)(x) into f(x) and set it equal to 12.
f(x) = 12g^-1(x) becomes f(x) = 12[(x - 1) / 2].

Simplify the equation:
f(x) = 12(x - 1) / 2.
f(x) = 6(x - 1).

Now, set f(x) = 6(x - 1) equal to 12:
6(x - 1) = 12.

Solve for x:
6x - 6 = 12.
6x = 18.
x = 3.

Therefore, the value of x that solves the equation f(x) = 12g^-1(x) is x = 3.

Remember to always double-check your work and make sure your solutions satisfy the original equations.