Question

The functions f and g are such that

f(x)=4x-1 and g(x)=x²+3

a) find f^-1 (x)

b) if we are told that fg(x) = 2gf(x) we can show that
ax²-bx-3=0
where a and b are integers.
work out the value of a and the value of b.

Answer 1

The inverse function of f(x) is f^-1(x) = (x + 1)/4. When given fg(x) = 2gf(x), by comparing the coefficients after simplifying the equation, we determine that a = 8 and b = 4.

Step-by-step explanation:

To find f-1(x), we need to solve for x in terms of y from the equation y = 4x - 1. Adding 1 to both sides gives y + 1 = 4x, and then dividing both sides by 4 gives (y + 1)/4 = x. Thus, f-1(x) = (x + 1)/4.

To address part b), let's first compute fg(x) and gf(x). We have fg(x) = f(g(x)) = f(x2 + 3) = 4(x2 + 3) - 1 and gf(x) = g(f(x)) = g(4x - 1) = (4x - 1)2 + 3. Since we are told that fg(x) = 2gf(x), we can set up an equation 4(x2 + 3) - 1 = 2((4x - 1)2 + 3) and solve for x to find the values of a and b that satisfy the quadratic equation ax2 - bx - 3 = 0.

By expanding and simplifying the equation, we observe that a = 8 and b = 4 satisfy the quadratic equation, which is derived from the equality fg(x) = 2gf(x).

Answer 2

a)
`f(x)^-1=(x+1)/4\\\\`

b) a=28 ; b=-16

Step-by-step explanation:

*For b only:

We know that

`fg(x)---> 4(x^2+3)-1\\2gf(x)---> 2[(4x-1)^2+3]\\\\`

From this we create an equation

`4(x^2+3)-1= 2[(4x-1)^2+3]`

We solve the equation

`4x^2+11=32x^2-16x+8\\4x^2+11-32x^2+16x-8=0\\28x^2-16-3=0`