Questionusing zeroes, intercepts, asymptote, Rational functions solve $f(x)=\frac{2 x-1}{x+3}$

Studdy Solution

STEP 1

1. $f(x)$ is a rational function of the form $\frac{P(x)}{Q(x)}$ , where $P(x) = 2x - 1$ and $Q(x) = x + 3$ .
2. The zeroes of $f(x)$ occur where the numerator is zero.
3. The vertical asymptotes of $f(x)$ occur where the denominator is zero.
4. The horizontal asymptote of $f(x)$ can be determined by comparing the degrees of the numerator and the denominator.
5. The y-intercept occurs where $x=0$ .
6. The x-intercept occurs where $f(x) = 0$ .

STEP 2

1. Find the zeroes of $f(x)$ by solving $P(x) = 2x - 1 = 0$ .
2. Determine the vertical asymptote by solving $Q(x) = x + 3 = 0$ .
3. Determine the horizontal asymptote by analyzing the degrees of $P(x)$ and $Q(x)$ .
4. Find the y-intercept by evaluating $f(0)$ .
5. Find the x-intercept by solving $f(x) = 0$ .

STEP 3

Find the zeroes of $f(x)$ by solving $P(x) = 2x - 1 = 0$ .
$2x - 1 = 0$
Solve for $x$ :
$x = \frac{1}{2}$

STEP 4

Determine the vertical asymptote by solving $Q(x) = x + 3 = 0$ .
$x + 3 = 0$
Solve for $x$ :
$x = -3$

STEP 5

Determine the horizontal asymptote by comparing the degrees of $P(x)$ and $Q(x)$ .
The degrees of both the numerator and the denominator are 1. Therefore, the horizontal asymptote is determined by the ratio of the leading coefficients.
$\text{Horizontal Asymptote:} \quad y = \frac{2}{1} = 2$

STEP 6

Find the y-intercept by evaluating $f(0)$ .
$f(0) = \frac{2(0) - 1}{0 + 3} = \frac{-1}{3}$
So, the y-intercept is:
$\left(0, -\frac{1}{3}\right)$

STEP 7

Find the x-intercept by solving $f(x) = 0$ .
$\frac{2x - 1}{x + 3} = 0$
This occurs when the numerator is zero:
$2x - 1 = 0$
We already solved this in Step 1, yielding:
$x = \frac{1}{2}$
So, the x-intercept is:
$\left(\frac{1}{2}, 0 \right)$
Solution Summary: - Zero of $f(x)$ : $x = \frac{1}{2}$ - Vertical Asymptote: $x = -3$ - Horizontal Asymptote: $y = 2$ - Y-intercept: $\left(0, -\frac{1}{3}\right)$ - X-intercept: $\left(\frac{1}{2}, 0\right)$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms