Math Problem Statement

Consider the function f(x) = (x + 1) / x. (a) Fill in the table of values for x between -5 and 5. (b) Are there any values of x that cannot be inputted into f? (c) Sketch a graph of f(x).

Solution

Let's analyze the problem.

Given Function:

f(x)=x+1xf(x) = \frac{x+1}{x}

(a) Fill in the table:

The values of yy for corresponding xx are calculated as follows: f(x)=x+1x=1+1xf(x) = \frac{x+1}{x} = 1 + \frac{1}{x}

  • For x=5x = -5, f(x)=1+15=10.2=0.8f(x) = 1 + \frac{1}{-5} = 1 - 0.2 = 0.8
  • For x=4x = -4, f(x)=1+14=10.25=0.75f(x) = 1 + \frac{1}{-4} = 1 - 0.25 = 0.75
  • For x=3x = -3, f(x)=1+13=10.330.67f(x) = 1 + \frac{1}{-3} = 1 - 0.33 \approx 0.67
  • For x=2x = -2, f(x)=1+12=10.5=0.5f(x) = 1 + \frac{1}{-2} = 1 - 0.5 = 0.5
  • For x=1x = -1, f(x)=1+11=11=0f(x) = 1 + \frac{1}{-1} = 1 - 1 = 0
  • For x=1x = 1, f(x)=1+11=1+1=2f(x) = 1 + \frac{1}{1} = 1 + 1 = 2
  • For x=2x = 2, f(x)=1+12=1+0.5=1.5f(x) = 1 + \frac{1}{2} = 1 + 0.5 = 1.5
  • For x=3x = 3, f(x)=1+13=1+0.331.33f(x) = 1 + \frac{1}{3} = 1 + 0.33 \approx 1.33
  • For x=4x = 4, f(x)=1+14=1+0.25=1.25f(x) = 1 + \frac{1}{4} = 1 + 0.25 = 1.25
  • For x=5x = 5, f(x)=1+15=1+0.2=1.2f(x) = 1 + \frac{1}{5} = 1 + 0.2 = 1.2

The values seem mostly correct, but for x=3x = -3, the calculated value is approximately 0.670.67, not 0.50.5, and for x=4x = -4, it should be 0.750.75 instead of 0.860.86.

(b) Are there any values of xx that cannot be inputted into ff?

Yes, x=0x = 0 cannot be inputted because it would result in division by zero, which is undefined. The function f(x)f(x) includes 1x\frac{1}{x}, and division by zero is not allowed.

(c) Sketch of the graph of f(x)f(x):

  • As x0x \to 0, the function approaches a vertical asymptote since f(x)f(x) tends toward \infty for x>0x > 0 and -\infty for x<0x < 0.
  • For large positive and negative values of xx, the function approaches 1, creating a horizontal asymptote at y=1y = 1.

I can help you sketch the graph based on these observations. Would you like me to generate the graph?


Do you want any further clarification or details?

Here are five related questions for deeper understanding:

  1. What happens to the function f(x)f(x) as xx approaches 0 from the positive side?
  2. What is the behavior of f(x)f(x) as xx \to \infty?
  3. How do vertical and horizontal asymptotes affect the shape of a graph?
  4. Can you identify points where the function is increasing or decreasing?
  5. How would you describe the domain and range of f(x)f(x)?

Tip: Asymptotes play a key role in understanding the behavior of rational functions like f(x)f(x), especially at extreme values of xx.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotes
Graphing
Domain and Range

Formulas

f(x) = (x + 1) / x
f(x) = 1 + 1/x

Theorems

Division by Zero Undefined
Vertical Asymptotes
Horizontal Asymptotes

Suitable Grade Level

Grades 9-12