Problem 77 - Precalculus with Limits

Question

Think About It In Exercises \(75-78\) , use properties of exponents to determine which functions (if any) are the same. $$\begin{aligned} f(x) &=16\left(4^{-x}\right) \\ g(x) &=\left(\frac{1}{4}\right)^{x-2} \\ h(x) &=16\left(2^{-2 x}\right) \end{aligned}$$

Step-by-Step Solution

Answer

After simplifying and comparing the functions f(x), g(x) and h(x), it can be concluded that these functions are the same as they all simplify to \(16(2^{-2x})\).

Step 1: Simplify Function f(x)

Begin by simplifying the function \(f(x) = 16(4^{-x})\) using the property that \(a^{-n} = 1/(a^{n})\). So the equation simplifies to \(f(x) = 16/(4^{x})\). Now, replace \(4\) as \(2^2\), then the function \(f(x)\) simplifies further to \(f(x) = 16/(2^{2x})\).

Step 2: Simplify Function g(x)

Now, simplify the function \(g(x) = (1/4)^{x-2}\). We know that \((1/a)^n = a^{-n}\), so the function simplifies to \(g(x) = 4^{-x+2} = 4^{2-x}\). Replacing \(4\) as \(2^2\), we get \(g(x) = (2^{2})^{2-x} = 2^{4-2x}\). Multiply top and bottom by \(16\) and then this function transforms to \(g(x) = 16(2^{-2x})\).

Step 3: Simplify Function h(x)

The function \(h(x) = 16(2^{-2x})\) is already in a simplified form.

Step 4: Comparison of f(x), g(x), and h(x)

After simplifying \(f(x)\), \(g(x)\), and \(h(x)\), it's clear to note that \(f(x) = g(x) = h(x) = 16(2^{-2x})\). Therefore, the functions are indeed the same.

Key Concepts

Simplifying Expressions

When working with mathematical expressions, especially those involving exponents, simplifying them can create a clearer path to understanding and solving problems. Understanding the properties of exponents is fundamental to simplifying expressions. Common properties include the negative exponent rule \( a^{-n} = \frac{1}{a^n} \) and the power of a power rule \( (a^m)^n = a^{mn} \).

For example, the expression \( 4^{-x} \) can be rewritten using the negative exponent rule as \( \frac{1}{4^x} \). Further simplification might involve expressing the base 4 as \( 2^2 \) since exponentiation is based on the understanding of powers and can be simplified accordingly using the rules of exponents we mentioned. Simplification often involves making substitutions or transforming the expression to reveal some sort of equivalence or to make the expression easier to work with.

Exponential Functions

Exponential functions are a type of mathematical function characterized by an equation where the variable appears as an exponent. A basic form of an exponential function is \( f(x) = b^x \), where \( b \) is the base and \( x \) is the exponent. Exponential functions grow or decay at rates proportional to their current value, which makes them particularly important in the realms of science, finance, and many other fields.

For instance, the functions in the given exercises are forms of exponential functions because they involve exponents that are variables. By using properties of exponents, one can rewrite these functions to reveal their true nature. It's significant to note how different representations of an exponential function can still result in the same graph or output values when simplified adequately, as is the case with the functions \( f(x) \), \( g(x) \), and \( h(x) \) in the original exercise.

Function Comparison

Comparing functions involves looking at the structure and outputs of different functions to determine any relationships or equivalences between them. When functions appear different, applying simplification techniques can reveal whether they produce the same result. This becomes clear through algebraic manipulation which includes expanding, factoring, and applying exponent rules.

As demonstrated in the original exercise, functions \( f(x) \), \( g(x) \), and \( h(x) \) were compared post-simplification to verify that they are indeed the same. They're equivalent because they all produce the same value for any input \( x \). This comparison highlights how understanding the properties of exponents is essential in recognizing equivalent expressions and thereby verifying if functions are the same or different. Function comparison not only assists in solving textbook exercises but also in grasping deeper mathematical relationships.