Math Problem Statement

Find the range and domain of the exponential function shown in the graph. The horizontal asymptote is given as a dashed line.

Solution

Based on the uploaded problem:

  • Graph Type: Exponential function.
  • Horizontal Asymptote: A dashed line is shown, which indicates the graph approaches but does not touch this line.

Step 1: Domain

The domain of an exponential function is all real numbers, because you can input any xx-value into the exponential function, and it will yield a corresponding yy-value.

  • Domain: x(,)x \in (-\infty, \infty) or "All reals."

Step 2: Range

The graph of an exponential function typically approaches a horizontal asymptote but never crosses it. Based on the graph:

  • The horizontal asymptote is at y=0y = 0, meaning the function never reaches or dips below y=0y = 0.

  • The yy-values are strictly greater than 0 (y>0y > 0).

  • Range: y>0y > 0.

Final Answer:

(a) Range: y>0y > 0
(b) Domain: x(,)x \in (-\infty, \infty)

Would you like further clarification or examples of similar problems?
Here are 5 related questions for further exploration:

  1. How would the graph change if the asymptote moved to y=2y = -2?
  2. What is the range and domain of a logarithmic function?
  3. How do transformations (shifts, stretches, reflections) affect the domain and range of an exponential function?
  4. How do we identify the asymptote of an exponential function algebraically?
  5. Can an exponential function have a range that includes negative values?

Tip: Always verify the asymptote and direction of the graph before determining range.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Domain and Range
Asymptotes

Formulas

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Theorems

Exponential Function Properties
Range and Domain of Functions

Suitable Grade Level

Grades 9-12