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 45

Chapter 5, Problem 45

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^2x + 1

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 inside the exponent for $h(x) = e^{2x} + 1$. The \$2x$ means a horizontal compression by a factor of \(\frac{1}{2}$ compared to \)f(x)$ because the input $x$ is multiplied by 2.

The $+1$ outside the exponential function shifts the entire graph vertically upward by 1 unit. This also shifts the horizontal asymptote from $y = 0$ to $y = 1$.

Write the equation of the asymptote explicitly: $y = 1$. This is the new horizontal asymptote for $h(x)$.

Determine the domain and range of $h(x)$. Since the exponential function is defined for all real numbers, the domain remains $(-\infty, \infty)$. The range shifts up by 1, so the new range is $(1, \infty)$.

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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 a horizontal asymptote at y = 0. Understanding this base graph is essential for applying transformations.

Transformations of Functions

Transformations include shifts, stretches, compressions, and reflections applied to the base graph. For h(x) = e^(2x) + 1, the exponent 2x compresses the graph horizontally, and the +1 shifts it upward by 1 unit, changing the asymptote from y=0 to y=1. Recognizing these changes helps in sketching and analyzing the new graph.

Domain, Range, and Asymptotes of Exponential Functions

The domain of exponential functions is all real numbers, while the range depends on vertical shifts and reflections. The horizontal asymptote is a line the graph approaches but never touches, often y=0 for e^x. For h(x) = e^(2x) + 1, the asymptote shifts to y=1, and the range becomes (1, ∞).

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Domain & Range of Transformed Functions

Related Practice

Textbook Question

Graph f(x) = (1/2)^x and g(x) = log_1/2 x in the same rectangular coordinate system.

Textbook Question

Graph f(x) = 4^x and g(x) = log₄ x in the same rectangular coordinate system.

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. e^4x+5e^2x−24=0

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log₂ (96) - log₂ (3)

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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 g(x) = 2e^x

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