Algebra 2...help please f(x)= x^2 / 4.?

Question

So basically I was sick for a day and missed out on a new Lesson. The teacher told me to look in the book for help and get the notes from fellow students.

So anyways We are learing Inverses of Function -- The Logarithmic Funtion

Can someone help me find the Inverse of f(x)= x^2 / 4.

I know for a fact that the given equation will be y = x^2 / 4
and when you inverse it, it will be x = y^2 / 4.

If someone could tell me how you can solve for y in the inverse equation (with some steps shown please!) that would be really great.

please and thank you.

Answer 1

If y = x^2 / 4 then multiply both sides by 4 and get 4y = x^2. Now take the square root of both sides and get x = square root of 4y or
x = -(square root of 4y). But the square root of 4y is 2*(square root of y). So the inverse of y = x^2 / 4 is actually x = 2(square root of y) or x = -2(square root of y)

Answer 2

There's a lot to talk about here, luv, but I'll try to summarize it.

First off, every function has an inverse (although they aren't always easy to find), but not every function has an inverse function. The key is the word "function" - in other words, is the inverse a function or not?

For instance, consider the function

f(x) = 3x + 4

The inverse would be

f-1(x) = (x - 4)/3

This is a function, so it's an inverse function.

On the other hand, consider:

f(x) = x^2

The inverse would be

f-1(x) = +/- sqrt(x)

This is NOT a function - the key being the whole "plus or minus" thing. So x^2 has an inverse but no inverse function.

Now: What you appear to be doing to find the inverse is to rewrite the function in terms of y and x, then switch them, and solve for y. This will work, but you'll realize pretty soon that there's no point to it. Consider:

f(x) = 5x - 10

y = 5x - 10

x = 5y - 10

5y = x + 10

y = x/5 + 2

f-1(x) = x/5 + 2

This is the correct inverse function, but that whole bit about switching the x and y is really just wasting time. It might be a good idea to do it for awhile, but eventually you're just going to skip it.

Now, for your specific example:

f(x) = (x^2)/4

Let's follow your pattern to find the inverse.

y = (x^2)/4

x = (y^2)/4

y^2 = 4x

y = +/-sqrt(4x)

y = +/- 2 sqrt(x)

I will not write this as f-1(x) because it's not a function. We've got the whole "plus/minus" problem again.

There's a lot more I could say about this, but I think I've covered what you needed. I hope it helped.

Edit: Corrected minor stupidity.

Answer 3

x = y^2 / 4
4x=y^2
(4x)^(1/2)=(y^2)^(1/2)
2x^(1/2)=y