A few explanations about combining functions

[JSmith]

What type of function results from the composition of two linear functions? Explain.
not sure how to answer this one... the composition of two linear functions results in another linear function. anything else to throw in?

Why is (a, a) a point on f^-1(f(x)) if (a, b) is a point on f(x)? Explain.
I remember hearing that as long as f(x) and its inverse are functions, the combination of the two always produces x. This would explain why if you input a, a would be the output as well. Is this right?

 

[soroban]

Hello, JSmith!

What type of function results from the composition of two linear functions? Explain.

One linear function is: .\(\displaystyle f(x) \:=\:ax + b\)
Another linear function is: .\(\displaystyle g(x) \:=\:cx + d\)

Then: .\(\displaystyle (f\circ g)(x) \:=\:a(cx+d) + b \:=\:acx + ad + b\) . . . a linear function.

And: . \(\displaystyle (g\circ f)(x) \:=\:c(ax+b) + d \:=\:acx + bc + d\) . . . a linear function.

 

[Deleted member 4993]

What type of function results from the composition of two linear functions? Explain.
not sure how to answer this one... the composition of two linear functions results in another linear function. anything else to throw in?

Why is (a, a) a point on f^-1(f(x)) if (a, b) is a point on f(x)? Explain.
I remember hearing that as long as f(x) and its inverse are functions, the combination of the two always produces x. This would explain why if you input a, a would be the output as well. Is this right? correct

f^-1(f(x)) = x → f^-1(f(a)) = a