Question

Determine whether each pair of functions are inverse functions. Write yes or no.

(Fog)(x) and (gof)(x)



f(x) = 2 squarroot x – 5
g(x) = 1/4x^2 - 5

Answer 1

The given functions f(x) and g9x) are inverse functions

How to find if the pair of functions are inverse functions

The functions to be worked are

f(x) = 2√(x + 5)

g(x) = 1/4 x² - 5

Let us solve for invers of f(x) = 2√(x + 5

say y = f(x) = 2√(x + 5

make x the subject of formula

y = 2√(x + 5)

y/2 = √(x + 5)

(x + 5) = (y/2)²

x = y²/4 - 5

interchanging the variables

y = x²/4 - 5

y = 1/4 x² - 5 = g(x), hence they are inverse functions

Answer 2

Answer: No

Explanation:

There are two ways to solve this. The first way is to calculate g(f(x)) (or f(g(x))). If f and g are inverse functions f(g(x)) = g(f(x)) = x.

g(f(x)) = (1/4)(2sqrt(x - 5))^2 - 5

= (1/4)*2^2*sqrt(x - 5)^2 - 5

= (1/4)*4*(x - 5) - 5

= x - 5 - 5

= x - 10

g(f(x)) = x - 10 =/= x, so the two functions are not inverse functions.

You can also solve this by using a test value a and calculating g(f(a)). If f and g are inverse functions, the result will be a.

For example, let a = 6.

f(6) = 2*sqrt(6 - 5)

= 2*sqrt(1)

= 2*1

= 1

g(f(6)) = g(1)

= (1/4)*1^2 - 5

= (1/4)*1 - 5

= 1/4 - 5

= -4 3/4

=/= 6

Therefore f and g are not inverse functions

Please note that this technique does not work in the other direction. If g(f(a)) = a, this does NOT prove that f and g are inverse functions.