Question

Use the following functions for questions 5 and 6.
f(x)= x^2-7x+3 and g(x)=x-2
5. Determine (f∘g)(x). 6. Determine (f ∘g)(5).

Answer 1

Find the inverse of f(x)= -1/5x+1.

Answer 2

To determine (f∘g)(x), we need to substitute g(x) into f(x).

(f∘g)(x) = f(g(x))

Substituting g(x) = x-2 into f(x):

(f∘g)(x) = f(x-2)

We can now substitute x-2 into f(x) to find the composition:

(f∘g)(x) = (x-2)^2 - 7(x-2) + 3

Expanding this expression:

(f∘g)(x) = x^2 - 4x + 4 - 7x + 14 + 3

Combining like terms:

(f∘g)(x) = x^2 - 11x + 21

Therefore, (f∘g)(x) = x^2 - 11x + 21.

To determine (f ∘g)(5), we substitute x = 5 into (f∘g)(x):

(f∘g)(5) = (5)^2 - 11(5) + 21

Calculating this expression:

(f∘g)(5) = 25 - 55 + 21

(f∘g)(5) = -9

Therefore, (f ∘g)(5) = -9.

Answer 3

To find the inverse of a function, we need to swap the roles of x and y and solve for y.

Let's start by swapping x and y:

x = -1/5y + 1

Next, we'll solve for y:

x - 1 = -1/5y (subtract 1 from both sides of the equation)

-5(x - 1) = y (multiply both sides by -5 to isolate y)

-5x + 5 = y

Now, we can rewrite the equation with y instead of x:

y = -5x + 5

Therefore, the inverse of f(x) = -1/5x + 1 is f^(-1)(x) = -5x + 5.