Question 2
Find all intercept(s) of the function: .
Question 3
Find all vertical asymptote(s) of the function .

Ask by Owen Goodwin. in the United States

Mar 19,2025

Question

Question 2 Find all intercept(s) of the function: $$ R(x)=\frac{2 x-3}{5 x^{2}+2 x-3} $$. Question 3 Find all vertical asymptote(s) of the function $$ q(x)=\frac{2 x-3}{5 x^{2}+2 x-3} $$.

UpStudy AI · Accepted Answer

To find the intercepts of the function $$ R(x) = \frac{2x-3}{5x^2+2x-3} $$, we need to find the values of $$ x $$ where the function intersects the x-axis. This means that the function will be equal to zero at these points.

To find the vertical asymptotes of the function $$ q(x) = \frac{2x-3}{5x^2+2x-3} $$, we need to find the values of $$ x $$ where the denominator of the function is equal to zero. These values will give us the vertical asymptotes of the function.

Let's start by finding the intercepts of the function $$ R(x) $$ and then the vertical asymptotes of the function $$ q(x) $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2x-3=0$$
- step1: Move the constant to the right side:
  $$2x=0+3$$
- step2: Remove 0:
  $$2x=3$$
- step3: Divide both sides:
  $$\frac{2x}{2}=\frac{3}{2}$$
- step4: Divide the numbers:
  $$x=\frac{3}{2}$$
The intercepts of the function $$ R(x) = \frac{2x-3}{5x^2+2x-3} $$ are $$ x = \frac{3}{2} $$ and $$ x = 1.5 $$.

Now, let's find the vertical asymptotes of the function $$ q(x) = \frac{2x-3}{5x^2+2x-3} $$.
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
  $$5x^{2}+2x-3=0$$
- step1: Factor the expression:
  $$\left(x+1ight)\left(5x-3ight)=0$$
- step2: Separate into possible cases:
  $$\begin{align}&5x-3=0\&x+1=0\end{align}$$
- step3: Solve the equation:
  $$\begin{align}&x=\frac{3}{5}\&x=-1\end{align}$$
- step4: Rewrite:
  $$x_{1}=-1,x_{2}=\frac{3}{5}$$
The vertical asymptotes of the function $$ q(x) = \frac{2x-3}{5x^2+2x-3} $$ are $$ x = -1 $$ and $$ x = 0.6 $$.

Therefore, the intercepts of the function $$ R(x) $$ are $$ x = \frac{3}{2} $$ and $$ x = 1.5 $$, and the vertical asymptotes of the function $$ q(x) $$ are $$ x = -1 $$ and $$ x = 0.6 $$.
The intercepts of $$ R(x) $$ are at $$ x = \frac{3}{2} $$ and $$ x = 1.5 $$. The vertical asymptotes of $$ q(x) $$ are at $$ x = -1 $$ and $$ x = 0.6 $$.

Question 2
Find all intercept(s) of the function: .
Question 3
Find all vertical asymptote(s) of the function .

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Question 2 Find all intercept(s) of the function: . Question 3 Find all vertical asymptote(s) of the function .