Algebra question, please help!?

Question

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

If f(X)= 1/2, what is x? what points are on the graph of f?

what is the domain of f?

List any x intercepts and y intercepts if there is one on the graph of f.

I know some of you probably think that I am making others do my homework, but that's not the case. I want to understand how to do this problem. It's been a long time since I took algebra. Thanks.

P.S. The person who thoroughly explains the answer wins and please show the steps.

Answer 1

equate the two statements. i.e:

(x^2+2)/x+4 = 1/2  to find x

Answer 2

Firstly, we know "f(x)=" is a fancy way of saying "y=". Also think f(x) = [a formula to change x] = y ... or "for every number x, something happens to that number, and the result is equal to y".

We have: f(x)= (x²+2) / (x+4)

If the result f(x)= 1/2, then we equate them:

1/2 = (x²+2) / (x+4)

We can solve for x like this:

(x+4)(1/2) = (x²+2)

(x+4)/2 = (x²+2)

(x+4) = 2(x²+2)

x+4 = 2x²+4

0 = 2x² - x

0 = x(2x-1) **(see note)**

Either x=0 or (2x-1)=0

So x=0 or x=1/2

**(note)**

The important thing is to factor! A common mistake would be to do this:

0 = 2x² - x

x = 2x²

1 = 2x

x = 1/2

Because you divided by x, you assume x was not 0 (you can't divide a number by 0, otherwise it would be undefined). But turns out that x COULD equal 0. By doing this you lost an answer!

*********

To find the other points of this function, use other numbers as x. We already know now that x can either be 0 or 1/2 to get 1/2. We got the points (0, 1/2) and (1/2, 1/2).

For example, if we were to let x=1, we plug this into the equation:

f(1) = (1²+2)/(1+4)

f(1) = (1+2)/(1+4)

f(1) = 3/5

We would get another point (1, 3/5). We keep doing this for as many x values we could to get a good picture of the graph. In fact the graph itself would look like this: http://i33.tinypic.com/20gxk7q.jpg

You can use a graphing calculator or an online Function Grapher like I did.

See how the graph supports the answers we got so far? =)

Also note the vertical line is not really part of the graph. It is called an asymptote, showing up where the resulting y value was undefined when x was -4.

But here's how to find the x and y intercepts without graphing. The x intercept is a point when the graph's line crosses the x-axis, or when the y value is 0. The y intercept when the graph crosses the y-axis or when the x value is 0.

First, with the x intercept, when y is equal to 0. We replace y or f(x) as 0:

0 = (x²+2)/(x+4)

We already have a problem. We don't want to multiply by (x+4), we might lose answers again ... But even if we continued:

0 = x²+2

-2 = x²

The square of a number can't be negative! It turns out there is no solution here. And, if you look at the image of our graph, no x-intercept! (Remember the asymptote is not part of the graph).

Now for the y-intercept, when x is equal to 0:

f(0) = (0²+2)/(0+4)

f(0) = 2/4 = 1/2

The y-intercept is 1/2! Or the point (0, 1/2)!

And that's how you do it.