Math Problem Statement

a and b and c are Real numbers and f function IS drawn on thé graph f(x)=a/1 +b/x+1 +c/x-3 now make f(x) function table

a and b and c are real numbers and the function f(x) is drawn on the graph where f(x) = a / 1 + b / (x + 1) + c / (x - 3). Now create a function table for f(x).

Solution

The function provided is:

f(x)=a1+bx+1+cx3f(x) = \frac{a}{1} + \frac{b}{x+1} + \frac{c}{x-3}

Based on the graph, we can analyze the behavior of f(x)f(x) at various intervals and determine approximate values for aa, bb, and cc by observing specific points and asymptotes. Here's the breakdown:

Observing Asymptotes and Points

  1. Asymptotes: The vertical lines x=1x = -1 and x=3x = 3 indicate vertical asymptotes, matching the terms bx+1\frac{b}{x+1} and cx3\frac{c}{x-3} in the function. This suggests that:

    • bb affects the behavior near x=1x = -1.
    • cc affects the behavior near x=3x = 3.
  2. Point Intersections: The graph shows intersections at specific points (e.g., points BB, CC, AA, and TT) that can help deduce values for aa, bb, and cc.

I will construct a function table by using values around key points and asymptotes.

Function Table Construction

For simplicity, I’ll use key values of xx near and away from the asymptotes:

xxf(x)f(x)
-\inftyApproaches a constant
1-1^-Approaches -\infty (asymptote)
1+-1^+Approaches ++\infty (asymptote)
00f(0)=a1+b+c3f(0) = \frac{a}{1} + b + \frac{c}{-3}
11f(1)=a1+b2+c2f(1) = \frac{a}{1} + \frac{b}{2} + \frac{c}{-2}
33^-Approaches -\infty (asymptote)
3+3^+Approaches ++\infty (asymptote)
++\inftyApproaches a constant

To create an exact function table, we would need the specific values of aa, bb, and cc. If you have any additional values from the graph or specific points, I can refine this table further.

Would you like more details on how to interpret values from the graph, or help with a specific calculation?

Additional Questions to Explore

  1. Can we calculate exact values of aa, bb, and cc from given points?
  2. What is the behavior of f(x)f(x) as x±x \to \pm \infty?
  3. How do the asymptotes affect the overall shape of f(x)f(x)?
  4. Are there any specific intersections that could further refine the values?
  5. How would adjusting aa, bb, and cc shift the function’s behavior?

Tip

Use the intersections and asymptotes in the graph to estimate values of constants by substituting known xx-values into f(x)f(x) and solving for unknowns.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotes
Function Behavior at Points
Function Intervals

Formulas

f(x) = a + b / (x + 1) + c / (x - 3)

Theorems

Asymptotic Behavior of Rational Functions
Limits and Intervals in Rational Functions

Suitable Grade Level

Grades 11-12