Math Problem Statement
Graph the function f(x) = e^{-(x-5)} if x < 5 and e^{x-5} if x ≥ 5. Based on the graph, state the domain, range, and find any intercepts.
Solution
The given problem involves analyzing a piecewise function:
\begin{cases} e^{-(x-5)} & \text{if } x < 5 \\ e^{x-5} & \text{if } x \geq 5 \end{cases}$$ ### 1. Domain: The domain of the function consists of all possible values of $$x$$ for which the function is defined. Since both pieces of the function are defined for all real numbers, the domain is: $$(-\infty, \infty)$$ ### 2. Range: To find the range, observe the behavior of each piece: - For $$x < 5$$, $$f(x) = e^{-(x-5)}$$. As $$x$$ approaches negative infinity, $$e^{-(x-5)}$$ becomes very large, approaching infinity. As $$x$$ approaches 5 from the left, $$e^{-(x-5)}$$ approaches 1. - For $$x \geq 5$$, $$f(x) = e^{x-5}$$. At $$x = 5$$, $$f(5) = 1$$. As $$x$$ increases, $$e^{x-5}$$ approaches infinity. Therefore, the range is: $$(0, \infty)$$ ### 3. Intercepts: - **Y-intercept**: To find the y-intercept, set $$x = 0$$: $$f(0) = e^{-(0-5)} = e^5$$ So, the y-intercept is $$(0, e^5)$$. - **X-intercept**: There is no x-intercept because $$f(x)$$ is always positive (since exponentials are never zero). ### Summary: - **Domain:** $$(-\infty, \infty)$$ - **Range:** $$(0, \infty)$$ - **Intercepts:** $$(0, e^5)$$ Would you like more details on any part? Here are five related questions to deepen understanding: 1. How would the domain and range change if the function were defined differently for $$x < 5$$? 2. What happens to the intercepts if a constant is added to $$f(x)$$? 3. Can you explain why exponentials are never zero? 4. How does the behavior of $$e^{-(x-5)}$$ compare with $$e^{x-5}$$? 5. How would the graph change if the piecewise conditions were modified? **Tip:** When analyzing a piecewise function, always examine each section independently to understand the overall behavior of the function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Exponential Functions
Domain and Range
Intercepts
Formulas
f(x) = e^{-(x-5)} for x < 5
f(x) = e^{x-5} for x ≥ 5
Theorems
Properties of Exponential Functions
Definition of Domain and Range
Intercepts of a function
Suitable Grade Level
Grades 10-12
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