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 39

Chapter 5, Problem 39

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

Verified step by step guidance

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

Analyze the transformation inside the exponent: The function $h(x) = e^{x-1} + 2$ has $x-1$ inside the exponent, which represents a horizontal shift. Specifically, replacing $x$ by $x - 1$ shifts the graph of $f(x)$ to the right by 1 unit.

Analyze the transformation outside the exponent: The $+2$ outside the exponential function shifts the entire graph vertically upward by 2 units. This affects the horizontal asymptote, moving it from $y = 0$ to $y = 2$.

Write the equation of the asymptote: Since the graph is shifted up by 2, the horizontal asymptote is $y = 2$.

Determine the domain and range of $h(x)$: The domain remains all real numbers $(-\infty, \infty)$ because exponential functions are defined for all real $x$. The range is shifted up by 2, so it becomes $(2, \infty)$.

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Key Concepts

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

Exponential Function and Its Graph

The exponential function f(x) = e^x is a fundamental function where the variable is in the exponent. Its graph passes through (0,1), is always positive, and increases rapidly. Understanding its shape and behavior is essential for applying transformations and analyzing domain and range.

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the base graph. For h(x) = e^(x-1) + 2, the graph shifts right by 1 unit and up by 2 units. Recognizing these changes helps in sketching the new graph and identifying changes in asymptotes, domain, and range.

Asymptotes, Domain, and Range of Exponential Functions

Exponential functions have a horizontal asymptote, typically y=0 for e^x. Transformations shift this asymptote accordingly, e.g., y=2 for h(x). The domain of e^x is all real numbers, and the range is positive real numbers; transformations affect the range but not the domain.

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Domain & Range of Transformed Functions

Related Practice

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

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log$ $\left$( $\frac{10x^2 \sqrt[3]{1 - x}$}{7(x + 1)^2} $\right$)$

Textbook Question

Evaluate each expression without using a calculator. log₅ 5⁷

Textbook Question

Evaluate each expression without using a calculator. log₄ 1

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) = e^x+2

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