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kramer
3 years ago
13

25 POINTS AND BRAINLIEST FOR THESE!

Mathematics
2 answers:
Verizon [17]3 years ago
8 0

Answer:

Step-by-step explanation:

Hello,

For any function f which has an inverse function we can write

x=(f^{-1}of)(x)=(fof^{-1})(x)=f(f^{-1}(x))

This is why, in practice, to find the inverse of f we will consider f(x) = y and we will look for x as a function of y, so we switch x and y and solve for y. Let's do it.

<u>Step 1 - The function f(x) can be written as a variable.  </u>\boxed{y}=f(x)<u />

f(x) = y = 5x + 2

<u />

<u>Step 2 - switch the variables x <-> y</u>

x = 5y + 2

subtract 2 to both parts of the equation

<=> x - 2 =  5y + 2 - 2 = 5y

divide by 5 both parts of the equation

y=\dfrac{x-2}{5}

It means that the inverse of f is as below.

\boxed{ \ f^{-1}(x)=\dfrac{x-2}{5}\ }

<u>Step 3 - Find the inverse of g(x)</u>

We already found that the inverse of f is g, so the inverse of g is f.

Let's do it again.

g(x)=y=\dfrac{x-2}{5} \ \ \text{ switch x and y } \\ \\ x= \dfrac{y-2}{5} \ \ \text{ solve for y }\\ \\ y-2=5x \ \ \text{ mulitply by 5 both parts of the equation } \\ \\ y = 5x+2 \ \ \text{ add 2 to both parts of the equation }

And we found what we already known, meaning f is the inverse of g.

(gof)(x)=(fog)(x)=x

Hope this helps.

Do not hesitate if you need further explanation.

Thank you

padilas [110]3 years ago
4 0

Answers and Step-by-step explanation:

<u>Step 1</u>:

We want to find the variable that ff(x) represents. Well, we know it can't be x because we already have x on the other side of the equation: ff(x) = 5x + 2.

So, ff(x) must equal y.

Since ff(x) = y, we know then that ff(x) = y = 5x + 2. And our equation is:

y = 5x + 2

<u>Step 2</u>:

Let's switch the variables now. This means that what used to be y will be x and what used to be x will be y:

y = 5x + 2  ⇒  x = 5y + 2

Subtract 2 from both sides:

5y = x - 2

Divide by 5 from both sides:

y = (x - 2)/5

<u>Step 3</u>:

Let's find the inverse of g(x) by doing the exact same thing as we did with ff(x):

g(x) = y = (x - 2)/5

Switch the variables:

y = (x - 2)/5  ⇒      x = (y - 2)/5

Multiply by 5 on both sides:

5x = y - 2

Add 2 to both sides:

y = 5x + 2

Notice that this is the exact same as ff(x)! This means that ff(x) and g(x) are inverses.

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