Answer:
2a)
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 to represent that function you have there.
We want to swap x and y:
Now we want to solve for y.
Subtract 2 on both sides:
Multiply both sides by 3:
Distribute:
So they want us to name the inverse g(x).
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.
Distribute:
Simplify:
.
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:
Simplify:
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.
