What Determines the Range of a Rational Function Like f(x) = 2/(x - 3)?

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SUMMARY

The range of the rational function f(x) = 2/(x - 3) is all real numbers except for 0, as the function approaches but never reaches this value. The domain of the inverse function is indeed the range of the original function, confirming that the output values of f(x) do not include 0. To find the range effectively, one can utilize algebraic manipulation and graphical analysis, focusing on the asymptotic behavior around the vertical asymptote at x = 3.

PREREQUISITES
  • Understanding of rational functions and their properties
  • Familiarity with inverse functions and their domains
  • Basic graphing skills for visualizing function behavior
  • Knowledge of asymptotes and limits in calculus
NEXT STEPS
  • Study the properties of rational functions and their ranges
  • Learn about finding the domain and range of inverse functions
  • Explore graphing techniques for rational functions using tools like Desmos
  • Investigate the concept of asymptotes and their significance in function analysis
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Students studying algebra, mathematics educators, and anyone interested in understanding the behavior of rational functions and their ranges.

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Find the range of
f(x) = 2/(x - 3).

1. What exactly are we looking for when we say RANGE of a rational function?

2. Is the domain of the inverse the range of the given function?

3. What is the easiest way to find the range? Graphing?
 
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1. All the values a function assumes over its domain.

2. Let's see:

y = 2/(x - 3)

x - 3 = 2/y

x = 2/y + 3

y^(-1) = 2/x + 3

What conclusions may we draw? (look at the values 3 and 0 in both functions).

3. Depends on the function (in my opinion).
 
greg1313 said:
1. All the values a function assumes over its domain.

2. Let's see:

y = 2/(x - 3)

x - 3 = 2/y

x = 2/y + 3

y^(-1) = 2/x + 3

What conclusions may we draw? (look at the values 3 and 0 in both functions).

3. Depends on the function (in my opinion).

I do not understand your answer to question 1.
I also do know what conclusions we can draw based on the values of 3 and 0 in both functions.
 

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