$f(x)$ is a rational function given in both standard and factored forms. Let $f(x)=\frac{3 x^{2}-14 x-24}{4 x^{2}-11 x+7}=\frac{(x-6)(3 x+4)}{(x-1)(4 x-7)}$ Find: 1) the domain in interval notation Note: Use $-\infty 0$ for $-\infty, \infty$ for $\infty, \cup$ for union. 2) the $y$ intercept at the point 3) $x$ intercepts at the point(s) 4) Vertical asymptotes at $x=$ 5) Horizontal asymptote at $y=$ Submit Question
Solution
Step 1 ：Given the function $f(x)=\frac{3 x^{2}-14 x-24}{4 x^{2}-11 x+7}=\frac{(x-6)(3 x+4)}{(x-1)(4 x-7)}$, we need to find the domain, the $y$ intercept, $x$ intercepts, vertical asymptotes and horizontal asymptote.
Step 2 ：The domain of a rational function is all real numbers except for the values of $x$ that make the denominator equal to zero. So, we need to find the values of $x$ that make the denominator $4x^2 - 11x + 7 = 0$.
Step 3 ：Solving the equation $4x^2 - 11x + 7 = 0$, we get $x = 1$ and $x = \frac{7}{4}$. These are the values of $x$ that make the denominator of the function equal to zero, so they are not included in the domain of the function.
Step 4 ：Therefore, the domain of the function is $(-\infty, 1) \cup (1, \frac{7}{4}) \cup (\frac{7}{4}, \infty)$.
Step 5 ：Final Answer: The domain of the function in interval notation is \(\boxed{(-\infty, 1) \cup (1, \frac{7}{4}) \cup (\frac{7}{4}, \infty)}\).
