Question

Math help, will give brainliest and 5 stars :)

Answer 1

Answer:

2a) $g(x)=3x-6$

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 $y=\frac{1}{3}x+2$ to represent that function you have there.

We want to swap x and y:

$x=\frac{1}{3}y+2$

Now we want to solve for y.

Subtract 2 on both sides:

$x-2=\frac{1}{3}y$

Multiply both sides by 3:

$3(x-2)=y$

$y=3(x-2)$

Distribute:

$y=3x-6$

So they want us to name the inverse g(x).

$g(x)=3x-6$

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.

$\frac{1}{3}(3x-6)+2$

Distribute:

$\frac{3x}{3}-\frac{6}{3}+2$

Simplify:

$x-2+2$

$x-0$

$x$.

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:

$3(\frac{1}{3})x+3(2)-6$

Simplify:

$1x+6-6$

$x+6-6$

$x+0$

$x$

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.