Plot points between and beyond each​ x-intercept and vertical asymptote. Find the value of the function at the given value of x.

x

negative 6−6

negative 5−5

33

55

77

f left parenthesis x right parenthesis equals StartFraction 7 x Over x minus 4 EndFractionf(x)=7xx−4

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​(Simplify your​ answers.)

We are given the function \( f(x) = \frac{7x}{x - 4} \), and we are tasked with finding the value of the function at specific values of \( x \), as well as plotting points beyond the x-intercepts and vertical asymptotes.

### 1. Vertical Asymptote
A vertical asymptote occurs where the denominator is zero. For the given function \( f(x) = \frac{7x}{x - 4} \), the denominator \( x - 4 = 0 \) when \( x = 4 \). Thus, there is a vertical asymptote at \( x = 4 \).

### 2. X-Intercept
To find the x-intercept, set the numerator equal to zero:
\[
7x = 0
\]
\[
x = 0
\]
So, the x-intercept is at \( (0, 0) \).

### 3. Finding the Values of the Function
We are asked to compute the values of the function at the given points.

- \( f(-6) = \frac{7(-6)}{-6 - 4} = \frac{-42}{-10} = 4.2 \)
- \( f(-5) = \frac{7(-5)}{-5 - 4} = \frac{-35}{-9} \approx 3.89 \)
- \( f(3) = \frac{7(3)}{3 - 4} = \frac{21}{-1} = -21 \)
- \( f(5) = \frac{7(5)}{5 - 4} = \frac{35}{1} = 35 \)
- \( f(7) = \frac{7(7)}{7 - 4} = \frac{49}{3} \approx 16.33 \)

### 4. Summary of Values:
- \( f(-6) = 4.2 \)
- \( f(-5) \approx 3.89 \)
- \( f(3) = -21 \)
- \( f(5) = 35 \)
- \( f(7) \approx 16.33 \)

### Conclusion
The vertical asymptote is at \( x = 4 \), and the x-intercept is at \( (0, 0) \). The function values at the specified \( x \)-coordinates have been calculated.

Would you like to go over any part in more detail, or should I help you plot this function?

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Here are 5 related questions you can explore:
1. What happens to the function as \( x \) approaches the vertical asymptote?
2. How do you find horizontal asymptotes for rational functions?
3. How do x-intercepts relate to the numerator of a rational function?
4. How does the sign of the function change around its intercepts and asymptotes?
5. Can a rational function have more than one vertical asymptote?

**Tip:** When dealing with rational functions, always check for both vertical asymptotes and horizontal asymptotes to understand their long-term behavior.

Plot points between and beyond each x-intercept and vertical asymptote. Find the value of the function at the given value of x. f(x) = (7x)/(x-4) for x = -6, -5, 3, 5, 7.

Learn how to plot the rational function f(x) = (7x)/(x-4), identifying its vertical asymptote at x=4 and the x-intercept at (0,0). Detailed calculations for the function values at x = -6, -5, 3, 5, and 7 are provided. Suitable for Grades 10-12.

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