Math  /  Algebra

Questionf(x)=3x3x23x+2f(x)=\frac{3 x-3}{x^{2}-3 x+2}
Answer Attempt 1 out of 2
Horizontal Asymptote: y=y= \square No horizontal asymptote
Vertical Asymptote: x=x= \square No vertical asymptote xx-Intercept: \square , 0) No xx-intercept \qquad yy-Intercept: (0, \square ) No yy-intercept
Hole: \square \square No hole

Studdy Solution

STEP 1

What is this asking? Find all the key features of this rational function, like where it crosses the axes, where it explodes to infinity, and where it has holes. Watch out! Don't forget to simplify the function first!
It'll make finding the features way easier.

STEP 2

1. Simplify the function
2. Find the horizontal asymptote
3. Find the vertical asymptote
4. Find the x-intercept
5. Find the y-intercept
6. Find the hole

STEP 3

We can **factor** out a 33 from the numerator: 3x3=3(x1)3x - 3 = 3(x-1)

STEP 4

We can **factor** the denominator as well: x23x+2=(x1)(x2)x^2 - 3x + 2 = (x-1)(x-2)

STEP 5

Now, we can **simplify** the whole function: f(x)=3(x1)(x1)(x2)f(x) = \frac{3(x-1)}{(x-1)(x-2)} If xx is *not* equal to 11, we can divide the numerator and denominator by (x1)(x-1), which gives us: f(x)=3x2f(x) = \frac{3}{x-2} This simplified form is super helpful for finding those key features, but remember that the original function isn't defined at x=1x = 1, so something interesting might happen there!

STEP 6

The **degree** of the numerator (00) is less than the **degree** of the denominator (11).

STEP 7

When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is always y=0y = 0.
Boom!

STEP 8

Vertical asymptotes happen when the denominator is **zero** and the numerator isn't.
In our simplified function, the denominator is (x2)(x-2).

STEP 9

This is zero when x=2x = 2.
The numerator is 33, which isn't zero, so we have a vertical asymptote at x=2x = 2.
Awesome!

STEP 10

To find the x-intercept, we set y=f(x)=0y = f(x) = 0 and solve for xx: 0=3x20 = \frac{3}{x-2}

STEP 11

There's no value of xx that can make this fraction equal to zero.
So, there's no x-intercept!

STEP 12

To find the y-intercept, we set x=0x = 0 in the simplified function: f(0)=302f(0) = \frac{3}{0-2}

STEP 13

f(0)=32=32f(0) = \frac{3}{-2} = -\frac{3}{2} So, the y-intercept is (0,32)(0, -\frac{3}{2}).
Fantastic!

STEP 14

Remember how we cancelled out (x1)(x-1) earlier?
That's where the hole is!

STEP 15

The hole is at x=1x = 1.
To find the y-value, plug x=1x = 1 into the simplified function: 312=31=3\frac{3}{1-2} = \frac{3}{-1} = -3

STEP 16

So, the hole is at (1,3)(1, -3).
Perfect!

STEP 17

Horizontal Asymptote: y=0y=0 Vertical Asymptote: x=2x=2 x-Intercept: None y-Intercept: (0,32)(0, -\frac{3}{2}) Hole: (1,3)(1, -3)

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