Sure, let's walk through the expression step-by-step.
We are given the function:
[tex]\[ f(x) = \frac{8x + 1}{3x - 6} \][/tex]
To understand and interpret this function, follow these steps:
1. Identify the numerator and denominator:
- The numerator is: [tex]\(8x + 1\)[/tex]
- The denominator is: [tex]\(3x - 6\)[/tex]
2. Simplify the function:
Look to see if there are any common factors between the numerator and the denominator that you can simplify. In this case, there are no common factors to simplify further.
3. Understand the domain of the function:
The function [tex]\( f(x) \)[/tex] is undefined when the denominator is [tex]\(0\)[/tex]. Therefore, we need to find the values of [tex]\(x\)[/tex] which make the denominator zero:
[tex]\[
3x - 6 = 0
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
3x = 6
\][/tex]
[tex]\[
x = 2
\][/tex]
Hence, the function is undefined at [tex]\( x = 2 \)[/tex]. Thus, the domain of [tex]\( f(x) \)[/tex] is all real numbers except [tex]\(x \neq 2\)[/tex].
4. Analyze the function properties:
- Vertical Asymptote: Since the denominator becomes 0 at [tex]\(x = 2\)[/tex], there is a vertical asymptote at [tex]\(x = 2\)[/tex].
- Horizontal Asymptote: For large values of [tex]\( |x| \)[/tex], both [tex]\(8x + 1\)[/tex] and [tex]\(3x - 6\)[/tex] are dominated by the terms with [tex]\(x\)[/tex]. Comparing the leading coefficients (the coefficients of [tex]\(x\)[/tex] in the numerator and the denominator), we can determine:
[tex]\[
\lim_{{x \to \infty}} \frac{8x + 1}{3x - 6} = \frac{8}{3}
\][/tex]
Therefore, the horizontal asymptote is [tex]\( y = \frac{8}{3} \)[/tex].
5. Intercepts:
- Y-intercept: To find the y-intercept, set [tex]\(x = 0\)[/tex]:
[tex]\[
f(0) = \frac{8(0) + 1}{3(0) - 6} = \frac{1}{-6} = -\frac{1}{6}
\][/tex]
- X-intercept: To find the x-intercept, set the numerator [tex]\(8x + 1 = 0\)[/tex]:
[tex]\[
8x + 1 = 0
\][/tex]
[tex]\[
8x = -1
\][/tex]
[tex]\[
x = -\frac{1}{8}
\][/tex]
So, the x-intercept is [tex]\( x = -\frac{1}{8} \)[/tex].
By following these steps, we have analyzed and interpreted the function [tex]\( f(x) = \frac{8x + 1}{3x - 6} \)[/tex] in detail.
