Question

PLEASE!!! NEED HELP ASAP!!! 50 PTS AND BRAINLIEST!!!

Answer 1

1) Inverse function: $f^{-1}(x)=g(x)=\frac{x+4}{3}$

2) $f(1) = -1, g(-1) = 1$

3) $f(g(x))=g(f(x))=x$

Step-by-step explanation:

1)

In this first method, we want to find the inverse function of $f(x)$.

The original function is:

$f(x) = 3x-4$

We rewrite it as

$y=3x-4$

We swap the name of the variables:

$x=3y-4$

Adding +4 on both sides:

$x+4=3y-4+4\\x+4=3y$

And dividing by 3 on both sides:

$y=\frac{x+4}{3}$

which is identical to $g(x)$:

$g(x)=\frac{x+4}{3}$

2)

In this second method, we want to use the output of one function as input to the other function, and show that the output value is equal to the imput value.

We start using the function

$f(x)=3x-4$

We choose $x=1$. We find:

$f(1)=3(1)-4=3-4=-1$

Now we use this output value as input in the function

$g(x)=\frac{x+4}{3}$

Substituting $x=-1$,

$g(-1)=\frac{-1+4}{3}=\frac{3}{3}=1$

So, the final output value (1) is equal to the input value (1).

3)

Here we want to verify that the two are inverse functions by showing that

$f(g(x))=x$

We have

$f(x)=3x-4\\g(x)=\frac{x+4}{3}$

Substituting g(x) into f(x),

$f(g(x))=3g(x)-4 = 3(\frac{x+4}{3})-4=(x+4)-4=x$

Viceversa, we want to show that

$g(f(x))=x$

Substituting g(x) into f(x), we get

$g(f(x))=\frac{f(x)+4}{3}=\frac{(3x-4)+4}{3}=\frac{3x-4+4}{3}=\frac{3x}{3}=x$

So, the two functions are one the inverse of the other.

Learn more about inverse functions:

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