Use composition of functions to show that the functions f(x) = 5x + 7 and
g(x)= 1/5x-7/5 are inverse functions. That is, carefully show that (fog)(x)= x and (gof)(x)= x.
f(x)=5x+7
g(x)=(1/5)x-7/5=(x-7)/5
(f⁰g)(x)
= f(g(x))
= f((x-7)/5)
= 5((x-7)/5)+7
= x-7 + 7
= x
(g⁰f)(x)
= g(f(x))
= g(5x+7)
= ((5x+7)-7)/5
= 5x/5
= x
I do not know how your book displays the expression, but the first term of
g(x)= 1/5x-7/5
offers two possible interpretations, namely (1/5)x or 1/(5x). You might even have been able to solve the problem if it wasn't for the ambiguity.
Thank you! I got a ton more of these to do, so you helped greatly! Thanks!
To show that functions f(x) = 5x + 7 and g(x) = 1/5x - 7/5 are inverse functions, we need to demonstrate that (fog)(x) = x and (gof)(x) = x.
Let's start by computing (fog)(x):
(fog)(x) = f(g(x)) = f(1/5x - 7/5)
To evaluate f(1/5x - 7/5), substitute the expression 1/5x - 7/5 for x in the function f(x):
f(1/5x - 7/5) = 5(1/5x - 7/5) + 7
= (1x - 7) + 7
= 1x - 7 + 7
= 1x
= x
Thus, (fog)(x) = x.
Now let's compute (gof)(x):
(gof)(x) = g(f(x)) = g(5x + 7)
Similarly, substitute the expression 5x + 7 for x in the function g(x):
g(5x + 7) = 1/5(5x + 7) - 7/5
= (1x + 7/5) - 7/5
= 1x + 7/5 - 7/5
= 1x
= x
Therefore, (gof)(x) = x.
Since both (fog)(x) and (gof)(x) are equal to x, we can conclude that the functions f(x) = 5x + 7 and g(x) = 1/5x - 7/5 are inverse functions.
g(f(x))=1/5 (5x+7) -7/5
= x+7/5-7/5=x
You do the other.