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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 1
Chapter 5, Problem 1

Fill in the blank(s) to correctly complete each sentence. If ƒ(x) = 4x, then ƒ(2) = and ƒ(-2) = ________.

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1
Identify the given function: \(f(x) = 4^x\), which means the function raises 4 to the power of \(x\).
To find \(f(2)\), substitute \(x = 2\) into the function: \(f(2) = 4^{2}\).
Calculate the expression \$4^{2}$ by multiplying 4 by itself twice (though we won't compute the final value here).
To find \(f(-2)\), substitute \(x = -2\) into the function: \(f(-2) = 4^{-2}\).
Recall that a negative exponent means taking the reciprocal: \(4^{-2} = \frac{1}{4^{2}}\). This completes the expression for \(f(-2)\).

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Key Concepts

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

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant. The function grows or decays depending on the base, and the exponent x can be any real number. Understanding how to evaluate these functions at specific values of x is essential.
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Exponential Functions

Evaluating Functions at Given Inputs

To find f(c) for a function f(x), substitute the input value c into the function in place of x. This process involves performing the necessary arithmetic operations to simplify and find the output value corresponding to the input.
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Evaluating Composed Functions

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent, i.e., a^{-n} = 1 / a^n. This rule is crucial when evaluating exponential functions at negative inputs, ensuring correct calculation of values like f(-2).
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Zero and Negative Rules
Related Practice
Textbook Question

Solve each equation. Round answers to the nearest hundredth as needed. (1/4)x=64

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Textbook Question

Fill in the blank(s) to correctly complete each sentence. The graph of ƒ(x) = -(1/3)x+4-5 is that of ƒ(x) = (1/3)x reflected across the ______ -axis, translated to the left ______ units and down _______ units.

Textbook Question

Answer each of the following. Write log3 12 in terms of natural logarithms using the change-of-base theorem.