Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 37

Chapter 5, Problem 37

The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = e^x+2

Verified step by step guidance

Start with the base function $f(x) = e^{x}$, which is an exponential function with a horizontal asymptote at $y = 0$, domain $(-\infty, \infty)$, and range $(0, \infty)$.

Identify the transformation in the given function $g(x) = e^{x} + 2$. This represents a vertical shift of the graph of $f(x)$ upward by 2 units.

Apply the vertical shift to the graph: every point on the graph of $f(x)$ moves up 2 units, so the new graph of $g(x)$ will be the same shape but shifted upward.

Determine the new horizontal asymptote by shifting the original asymptote $y = 0$ up by 2 units, resulting in the asymptote $y = 2$ for $g(x)$.

State the domain and range of $g(x)$: the domain remains all real numbers $(-\infty, \infty)$, and the range shifts up by 2, becoming $(2, \infty)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Functions and Their Graphs

An exponential function has the form f(x) = a^x, where the base a is positive and not equal to 1. The graph of f(x) = e^x is a smooth curve increasing rapidly, passing through (0,1), with domain all real numbers and range (0, ∞). Understanding this base graph is essential before applying transformations.

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the base graph. For g(x) = e^x + 2, the graph shifts vertically upward by 2 units. Recognizing how adding constants affects the graph helps in sketching and identifying new asymptotes and ranges.

Asymptotes, Domain, and Range

An asymptote is a line the graph approaches but never touches. For f(x) = e^x, the horizontal asymptote is y = 0. Vertical shifts change the asymptote accordingly, so for g(x) = e^x + 2, the asymptote is y = 2. The domain remains all real numbers, while the range shifts to (2, ∞).

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Determining Horizontal Asymptotes

Related Practice

Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 7^(x+2)=410

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Textbook Question

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = log x and g(x) = - log (x+3)

Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\ln$ $\left$( $\frac{x^3 \sqrt{x^2 + 1}$}{(x + 1)^4} $\right$)$

Textbook Question

The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e^x-1+2

Textbook Question

Evaluate each expression without using a calculator. log₄ 1

Textbook Question

In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log2 (x-2)

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