Consider the functions f(x) and g(x)

Question

Consider the functions f(x)=-5x+1 and g(x)=-1/5(x-1)

a)find f(g(x))

b)find g(f(x))

c)determine whether the functions f and g are inverses of each other

d)give any values of x that need to be excluded.

Answer 1

To find f(g(x)take g(x) and substitute it for x in the function f(x)

so it is 5(1/(5x-1) +1 = (1/x+1) +1

In a similar manner to fine g(f(x) Take f(x) and substitute it for x in the function g(x)

This gives us 1/ 5(5x+1 -1)= 1/5(5x)= 1/25x

Since f(g(x) doe not equal g(f(x) they are not inverses

In the function g(x) you need to exclude x=1 from the domain since that would make the denominator undefined

Answer 2

a) f(g(x)) = -5 ( -1/5(x-1)) + 1

= x-1 + 1

= x

b) g(f(x)) = -1/5 ( -5x+1 - 1) = -1/5(-5x) = x

c) y = -5x +1 ----> x = -5y + 1

(x-1) = -5y

(x-1)/(-5) = y

(-1/5)(x-1)

Yes they are inverses as verified parts (a) and (b). Please note how the inverse is written using

parenthesis as appropriate

d) the domain is all reals for both the function and it's inverse. Note that the inverse of

a linear function is also a linear function, so the domain is all reals