Problem 76 - Precalculus with Limits

Question

Think About It In Exercises \(75-78\) , use properties of exponents to determine which functions (if any) are the same. $$\begin{array}{l}{f(x)=4^{x}+12} \\ {g(x)=2^{2 x+6}} \\\ {h(x)=64\left(4^{x}\right)}\end{array}$$

Step-by-Step Solution

Answer

The functions that are the same are \(g(x) = 2^{2 x+6}\) and \(h(x) = 64*4^{x}\).

Step 1: Simplify f(x)

The function \(f(x)\) is already in its simplest form, so there is no need to simplify it.

Step 2: Simplify g(x)

The function \(g(x) = 2^{2x+6}\) can be rewritten as \(2^{2x} * 2^{6}\), applying the rule \(a^{m+n} = a^m * a^n\). This results in \(g(x) = 4^x * 64\).

Step 3: Simplify h(x)

The function \(h(x) = 64*4^{x}\) is already in its simplest form.

Step 4: Compare the functions

From the simplification steps above, we see that \(g(x) = 4^x * 64\) is equivalent to \(h(x) = 64*4^{x}\), but they are not equivalent to \(f(x) = 4^{x}+12\).

Key Concepts

Exponential Functions

Exponential functions represent one of the fundamental concepts in mathematics and are used to describe growth patterns that increase at a constant rate. An exponential function is defined by an equation of the form \( f(x) = a^{bx} \), where \( a \) is a positive constant referred to as the base, \( b \) is the exponent, and \( x \) is the variable.

An important characteristic of exponential functions is their rapid growth or decay. When the base \( a \) is greater than 1, the function represents exponential growth, and when \( a \) is between 0 and 1, it represents exponential decay. Exponents play a key role in this as they determine how fast the function grows or decays with changes in \( x \).

In our given exercise, the functions \( f(x) = 4^{x}+12 \), \( g(x) = 2^{2x+6} \), and \( h(x) = 64\text{*}4^{x} \) all include exponential expressions. By understanding the properties of exponentials, such as \( a^{m} * a^{n} = a^{m+n} \) and \( (a^{b})^{c} = a^{bc} \), we can manipulate and compare these functions effectively.

Simplifying Expressions

Simplifying expressions is a key skill in mathematics, allowing students to transform complex equations into more manageable forms. This simplification often uses properties of exponents, such as the product of powers rule (\( a^{m} * a^{n} = a^{m+n} \)), the power of a product rule (\( (ab)^{n} = a^{n} * b^{n} \)), and the power of powers rule (\( (a^{m})^{n} = a^{mn} \)).

For example, to simplify the function \( g(x) = 2^{2x+6} \), we apply the product of powers rule to rewrite the exponential expression as \( 2^{2x} * 2^{6} \), which then becomes \( 4^{x} * 64 \), since \( 2^6 = 64 \) and \( 2^{2x} = (2^2)^x = 4^x \). Recognizing these properties allows students to break down and simplify functions considerably, making them easier to understand and compare.

Exercise Improvement Advice

When simplifying expressions, it's essential to be systematic. Start by identifying like terms and common factors, and make use of the exponent rules. It's also helpful to memorize basic exponent values for small powers, which can speed up the process of simplification.

Function Comparison

Comparing functions is a valuable skill that involves checking if two or more functions have the same value for all inputs, or identifying particular inputs where they may intersect or diverge. When dealing with exponential functions, we look for similarities or differences in bases and exponents.

For instance, after simplifying the expressions in our exercise, we found that \( g(x) = 4^{x} * 64 \) is the same as \( h(x) = 64*4^{x} \), indicating they are equivalent functions for all values of \( x \). However, \( f(x) = 4^{x}+12 \) differs due to the addition of 12; it cannot be simplified in a way that makes the exponential part of the expression factor out as with \( g(x) \) and \( h(x) \).

Therefore, a close study of the properties of the exponents and the structure of the functions can lead to efficient and accurate comparisons. This understanding not only helps in verifying whether functions are same or not but also aids in graphing and solving real-world problems that can be modeled using exponentials.