Answer
q1.
Given: [tex]y = \frac{9}{x^2}+2[/tex]
Rewrite this function in the way as;
[tex]y = \frac{9}{g(x)} + 2[/tex]
then;
[tex]g(x) = x^2[/tex] and [tex]f(x) = \frac{9}{x} + 2[/tex]
q2.
Given the function:
[tex]f(x) = 9x+3[/tex] and [tex]g(x) = \frac{x-3}{9}[/tex]
Confirm f and g are inverses by showing f(g(x)) =x and g(f(x)) =x
First show f(g(x)) =x
f(g(x)) = 9(g(x)) + 3
Substitute the function g(x) we have;
[tex]f(g(x)) = 9(\frac{x-3}{9}) + 3[/tex]
or
[tex]f(g(x)) =(x-3) + 3[/tex] = x -3 + 3 = x
f(g(x)) =x
Similarly show that: g(f(x)) =x
[tex]g(f(x)) = \frac{f(x)-3}{9}[/tex]
Substitute the value of function g(x) we have;
[tex]g(f(x)) =\frac{9x+3-3}{9}[/tex]
Simplify:
[tex]g(f(x)) =\frac{9x}{9}[/tex]
[tex]g(f(x)) =x[/tex] hence proved!
Therefore, f and g are inverse of each other after showing f(g(x)) =x and g(f(x)) =x
A composite function is the combination of two or more functions to create another function.
Given that:
[tex]y = \frac 9{x^2} + 2[/tex]
Replace [tex]x^2[/tex] with g(x)
[tex]y = \frac 9{g(x)} + 2[/tex]
This means that:
[tex]g(x) = x^2[/tex]
Replace sub function g(x) with x and y, with f(x).
So, we have:
[tex]f(x) = \frac 9{x} + 2[/tex]
Hence, the equation is:
[tex]y = f(g(x))[/tex]
Where:
[tex]f(x) = \frac 9{x} + 2[/tex] and [tex]g(x) = x^2[/tex]
Given that:
[tex]f(x) = 9x + 3[/tex]
[tex]g(x) = \frac{x-3}9[/tex]
f(g(x)) is calculated as follows:
[tex]f(x) = 9x + 3[/tex]
Replace x with g(x)
[tex]f(g(x)) = 9\times g(x) + 3[/tex]
Substitute [tex]g(x) = \frac{x-3}9[/tex]
[tex]f(g(x)) = 9\times \frac{x - 3}{9} + 3[/tex]
[tex]f(g(x)) = x - 3 + 3[/tex]
[tex]f(g(x)) = x[/tex]
Similarly:
[tex]g(x) = \frac{x-3}9[/tex]
Replace x with f(x)
[tex]g(f(x)) = \frac{f(x)-3}9[/tex]
Substitute [tex]f(x) = 9x + 3[/tex]
[tex]g(f(x)) = \frac{9x + 3 - 3}{9}[/tex]
[tex]g(f(x)) = \frac{9x}{9}[/tex]
[tex]g(f(x)) = x[/tex]
Hence:
[tex]f(g(x)) = g(f(x)) = x[/tex]
Read more about composite functions at:
brainly.com/question/8776301
