Answer:
2a) [tex]g(x)=3x-6[/tex]
2b) This is a show kind of answer. The showing of this in the explanation.
2c) I provided the graph. The lines should be a reflection through the y=x line. The points (a,b) on y=(1/3)x+2 or swapped to get the points (b,a) on y=3x-6. That is for, example the point (-3,1) is on y=(1/3)x+2 while (1,-3) is on y=3x-6.
Step-by-step explanation:
2a) The inverse of a function is you just swapping x and y around. You also almost always asked to remake y the subject after that though.
So anyways we have this equation [tex]y=\frac{1}{3}x+2[/tex] to represent that function you have there.
We want to swap x and y:
[tex]x=\frac{1}{3}y+2[/tex]
Now we want to solve for y.
Subtract 2 on both sides:
[tex]x-2=\frac{1}{3}y[/tex]
Multiply both sides by 3:
[tex]3(x-2)=y[/tex]
[tex]y=3(x-2)[/tex]
Distribute:
[tex]y=3x-6[/tex]
So they want us to name the inverse g(x).
[tex]g(x)=3x-6[/tex]
2b) We want to show by composition that these functions are inverses. That is we want to show f(g(x))=x and g(f(x))=x.
Let's do that:
f(g(x))
Replace g(x) with 3x-6 since g(x)=3x-6.
f(3x-6)
Replace the old input x with the new input (3x-6) in (1/3)x+2.
[tex]\frac{1}{3}(3x-6)+2[/tex]
Distribute:
[tex]\frac{3x}{3}-\frac{6}{3}+2[/tex]
Simplify:
[tex]x-2+2[/tex]
[tex]x-0[/tex]
[tex]x[/tex].
So we do have f(g(x))=x.
Now to show the other way:
g(f(x))
Replace f(x) with (1/3)x+2 since f(x)=(1/3)x+2.
g((1/3)x+2)
Replace the old input x with the new input (1/3)x+2 in 3x-6.
3((1/3)x+2)-6
Distribute:
[tex]3(\frac{1}{3})x+3(2)-6[/tex]
Simplify:
[tex]1x+6-6[/tex]
[tex]x+6-6[/tex]
[tex]x+0[/tex]
[tex]x[/tex]
So we do have g(f(x))=x.
We have confirmed that f and g are indeed inverses since f(g(x))=x and g(f(x))=x.
2c) Visually if two functions are inverses they should be reflections through the y=x line so that is what we should see since f and g are inverses.
I going to compare both equations to y=mx+b form to determine the y-intercept and the slope.
y=mx+b
y=(1/3)x+2 tells us the slope is 1/3 and the y-intercept is 2.
y=3x-6 tells us the slope is 3 and the y-intercept is -6.
I have color-coded the picture.
