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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 13
Chapter 5, Problem 13

For ƒ(x) = 3x and g(x)= (1/4)x find each of the following. Round answers to the nearest thousandth as needed. ƒ(-2)

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Identify the function given: \( f(x) = 3^x \). We need to find \( f(-2) \), which means substituting \( x = -2 \) into the function.
Substitute \( -2 \) for \( x \) in the function: \( f(-2) = 3^{-2} \).
Recall the property of exponents for negative powers: \( a^{-n} = \frac{1}{a^n} \). So, \( 3^{-2} = \frac{1}{3^2} \).
Calculate the denominator: \( 3^2 = 3 \times 3 = 9 \), so \( f(-2) = \frac{1}{9} \).
If needed, convert \( \frac{1}{9} \) to a decimal and round to the nearest thousandth.

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Key Concepts

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

Exponential Functions

Exponential functions have the form f(x) = a^x, where the base a is a positive constant. They model growth or decay processes and are evaluated by raising the base to the power of the input x. Understanding how to compute values for negative and positive exponents is essential.
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Evaluating Functions at Specific Inputs

Evaluating a function at a given input means substituting the input value into the function's formula and simplifying. For example, to find f(-2), replace x with -2 and calculate the result, paying attention to exponent rules, especially with negative exponents.
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Evaluating Composed Functions

Rounding Decimal Values

After calculating function values, answers often need to be rounded to a specified decimal place. Rounding to the nearest thousandth means keeping three digits after the decimal point and adjusting the last digit based on the following digit to ensure accuracy and clarity.
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Related Practice
Textbook Question

For ƒ(x) = 3x and g(x)= (1/4)x find each of the following. Round answers to the nearest thousandth as needed. See Example 1. ƒ(2)

Textbook Question

Determine whether each function graphed or defined is one-to-one.

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Textbook Question

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. (1/2)x = 5

Textbook Question

If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in logarithmic form, write it in exponential form. 34 = 81

Textbook Question

Find each value. If applicable, give an approximation to four decimal places. log 0.1

Textbook Question

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. 3x = 7