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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 20
Chapter 5, Problem 20

For ƒ(x) = 3x and g(x)= (1/4)x find each of the following. Round answers to the nearest thousandth as needed. ƒ(-5/2)

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Identify the function given: \( f(x) = 3^x \). We need to find \( f\left(-\frac{5}{2}\right) \).
Substitute \( x = -\frac{5}{2} \) into the function: \( f\left(-\frac{5}{2}\right) = 3^{-\frac{5}{2}} \).
Recall the property of exponents for negative powers: \( a^{-b} = \frac{1}{a^b} \). So rewrite the expression as \( \frac{1}{3^{\frac{5}{2}}} \).
Express the fractional exponent \( \frac{5}{2} \) as a root and power: \( 3^{\frac{5}{2}} = \left(3^{\frac{1}{2}}\right)^5 = (\sqrt{3})^5 \).
Evaluate or approximate \( (\sqrt{3})^5 \) and then take the reciprocal to find \( f\left(-\frac{5}{2}\right) \), rounding the final answer 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 fractional exponents is essential.
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Evaluating Functions at Specific Inputs

Evaluating a function means substituting a given input value into the function's formula and simplifying. For example, to find f(-5/2), replace x with -5/2 and calculate the result carefully, especially when dealing with fractional or negative exponents.
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Evaluating Composed Functions

Rounding Decimal Values

After calculating a function value, rounding to a specified decimal place ensures clarity and precision. Rounding to the nearest thousandth means keeping three digits after the decimal point and adjusting the last digit based on the next digit's value.
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Related Practice
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. 6(x+1) = 4(2x-1)

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. g(-3)

Textbook Question

Find each value. If applicable, give an approximation to four decimal places. log(387 ×\(\times\) 23)

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. ex2=100e^{x^2} = 100

Textbook Question

Solve each equation. x = log3 1/81

Textbook Question

Determine whether each function graphed or defined is one-to-one. y = -√(100 - x2)

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