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Chapter 4: Problem 54

Use transformations of the graph of \(y=e^{x}\) to graph the function. Write the domain and range in interval notation. (See Example 3\()\) $$n(x)=-e^{x}+4$$

Short Answer

Expert verified
Domain: \((-\infty, \infty)\), Range: \((-fty, 4)\).

Step by step solution

01

- Identify the base function

The base function is given as \(y = e^x\). This is the starting point before any transformations.
02

- Apply a vertical reflection

The function \(n(x) = -e^x + 4\) includes a negative sign in front of \(e^x\). This indicates a reflection across the x-axis. The graph of \(e^x\) is flipped upside-down.
03

- Vertical shift

Next, add 4 to the entire function which causes a vertical shift upwards by 4 units. So, \(y = -e^x\) moves up 4 units to become \(y = -e^x + 4\).
04

- Identify the domain

The domain of any exponential function \(y = e^x\) and its transformations is all real numbers. Therefore, the domain is \((-\infty, \infty)\).
05

- Identify the range

The range of the original function \(y = e^x\) is \((0, \infty)\). After reflecting over the x-axis, the range becomes \((-fty, 0)\). Finally, applying the vertical shift upwards by 4 units changes the range to \((4, -fty)\). However, since -\infty always goes to the lower bound, the range should be written as \((-fty, 4)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Reflection
A vertical reflection flips the graph of a function across the x-axis.
For the exponential function, the base function is given by \(y = e^x\).
When we apply a vertical reflection, we introduce a negative sign in front of the function, transforming it to \(-e^x\).
This causes every point on the graph to be reflected across the x-axis, so the exponential curve that increases exponentially in the positive direction becomes a curve that decreases in the negative direction.
Imagine the original curve increasing sharply to the right and gently flattening to the left; after the vertical reflection, it will decrease sharply to the right and gently approach the x-axis upwards from negative infinity.
Vertical Shift
A vertical shift moves the entire graph of a function up or down without changing its shape.
In the function \(n(x) = -e^x + 4\), the term \(+4\) indicates a vertical shift.
Adding 4 to the entire function moves every point on the graph of \(-e^x\) up by 4 units.
This means that instead of the graph being flipped across the x-axis and lying below it, it will lie 4 units above where it typically would after the reflection.
Therefore, the point (0, -1) on the graph of \(-e^x\) becomes (0, 3) on the graph of \(-e^x + 4\). Each point moves up uniformly by 4 units, maintaining the shape while adjusting the vertical position.
Domain and Range
The domain and range of a function tell us the input values (x-values) it can accept and the output values (y-values) it can produce.
The domain of the original exponential function \(y = e^x\) includes all real numbers, written as \((- \infty, \infty)\).
This is because you can input any real number into an exponential function.
The range of \(y = e^x\) is \( (0, \infty)\) because the output is always positive and increases without bound as x becomes larger.
After applying the vertical reflection to get \(-e^x\), the range changes to \((- \infty, 0)\) since reflecting over the x-axis flips all positive outputs to negative outputs.
When we then apply the vertical shift of 4 units upwards, the entire range elevates by 4, resulting in the final range of \((- \infty, 4)\).
Now, the graph produces values from \(- \infty\) up to, but not including, 4. The domain remains unchanged as \((- \infty, \infty)\).

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Most popular questions from this chapter

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9\. The population of the United States \(P(t)\) (in millions) since January 1,1900 , can be approximated by $$P(t)=\frac{725}{1+8.295 e^{-0.0165 t}}$$ where \(t\) is the number of years since January \(1,1900 .\) (See Example 6\()\) a. Evaluate \(P(0)\) and interpret its meaning in the context of this problem. b. Use the function to predict the U.S. population on January \(1,2020 .\) Round to the nearest million. c. Use the function to predict the U.S. population on January 1,2050 . d. Determine the year during which the U.S. population will reach 500 million. e. What value will the term \(\frac{8.295}{e^{0.0165 t}}\) approach as \(t \rightarrow \infty\) ? f. Determine the limiting value of \(P(t)\).
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