Prove that the two functions are inverse functions. 𝑓(𝑥) = 2𝑥 − 7 and 𝑔(𝑥) =𝑥+7/2

Answer 1

To prove that two functions are inverse functions of each other, we need to show that when one function is composed with the other, it yields the identity function.

Let's first find the composition of 𝑓(𝑥) and 𝑔(𝑥):

(𝑓∘𝑔)(𝑥) = 𝑓(𝑔(𝑥))
= 𝑓(𝑥+7/2)
= 2(𝑥+7/2) - 7
= 2𝑥 + 14/2 - 7
= 2𝑥 + 7 - 7
= 2𝑥

Now, we can see that the composition of 𝑓(𝑥) and 𝑔(𝑥) yields the identity function 2𝑥.

Similarly, let's find the composition of 𝑔(𝑥) and 𝑓(𝑥):

(𝑔∘𝑓)(𝑥) = 𝑔(𝑓(𝑥))
= 𝑔(2𝑥 - 7)
= (2𝑥 - 7) + 7/2
= 2𝑥 - 14/2 + 7/2
= 2𝑥 - 7/2
= 𝑥

Now, we can see that the composition of 𝑔(𝑥) and 𝑓(𝑥) yields the identity function 𝑥.

Since both compositions yield the identity function, 𝑓(𝑥) = 2𝑥 - 7 and 𝑔(𝑥) = 𝑥 + 7/2 are inverse functions of each other.