Problem 78 - Precalculus with Limits

Question

Think About It In Exercises \(75-78\) , use properties of exponents to determine which functions (if any) are the same. $$\begin{array}{l}{f(x)=e^{-x}+3} \\ {g(x)=e^{3-x}} \\\ {h(x)=-e^{x-3}}\end{array}$$

Step-by-Step Solution

Answer

The functions \(f(x) = e^{-x} + 3\), \(g(x) = e^{3-x}\), and \(h(x) = -e^{x-3}\) are not the same.

Step 1: Express the functions in the same form

First, simplify the exponents for all the given functions to attain the same form for each function. This includes changing negative exponents to positive and expressing subtraction in the exponent as division. For \(f(x)\), there is nothing to change. For \(g(x)\), \(e^{3-x}\) can be written as \(e^3/e^x\), and for \(h(x)\), \(-e^{x-3}\) can be written as \(-e^x/e^3\).

Step 2: Compare the functions

After expressing all functions in the same form, they are compared to see if there are any that are equivalent to each other. However, on inspection, none of the functions \(f(x)\), \(g(x)\) and \(h(x)\) appear to be the same.

Key Concepts

Function Comparison

When comparing functions, we look at their algebraic structures to see if they are equivalent. To do this, it's essential to simplify the functions to a common form. This means using properties of exponents to rewrite the equations in ways that can easily be compared.
Start by simplifying each function independently. For our exercise, we looked at the functions:

• \(f(x) = e^{-x} + 3\)
• \(g(x) = e^{3-x}\)
• \(h(x) = -e^{x-3}\)

To compare them efficiently, use exponent rules like expressing subtraction in the exponent as division. This means, for instance, transforming \(e^{3-x}\) into \(\frac{e^3}{e^x}\) and \(-e^{x-3}\) into \(-\frac{e^x}{e^3}\). Such transformations help us see equivalences more clearly.
Finally, when the functions are in the same algebraic form, inspect them closely. If their structures match entirely, they are equivalent. In our case, no two functions ended up being similar, showing each function was unique in its algebraic structure.

Negative Exponents

Negative exponents can sometimes be intimidating, but breaking them down is quite straightforward. A negative exponent signifies the reciprocal of the base raised to the corresponding positive exponent.
Consider the function \(f(x) = e^{-x}\). The negative exponent \(-x\) indicates the reciprocal of \(e^x\), which can be rewritten as \(\frac{1}{e^x}\). This transformation is crucial for simplifying mathematical expressions and solving equations.

• To convert a function with a negative exponent, remember the rule \(a^{-b} = \frac{1}{a^b}\).
• Applying this rule can simplify many exponential expressions and make them easier to analyze together with other functions.

Using this method helps in converting and comparing functions like \(g(x)\) and \(h(x)\) to a common form, making them easier to analyze and compare.

Simplifying Exponential Expressions

To further understand exponential functions, you need to grasp how to simplify complex exponentials. Simplification involves using standard rules to break down expressions into a form that is easier to comprehend or compare.
For the given functions, simplification involved changing forms by applying exponent properties. For instance:

• \(e^{3-x}\) was simplified to \(\frac{e^3}{e^x}\)
• \(-e^{x-3}\) was simplified to \(-\frac{e^x}{e^3}\)

Guidelines for simplifying exponential expressions include:

• Using the rule \(a^{b-c} = \frac{a^b}{a^c}\), which helps break down expressions with subtracted exponents into a rational form.
• Converting negative exponents using \(a^{-b} = \frac{1}{a^b}\)\, as discussed earlier.

By simplifying the functions into a standard form, you can make more informed comparisons and conclusions. This strategy is key when your goal is to determine if functions are equivalent.