Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Use properties of exponents to write each function in the form \(f(t)=k a^{t},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\).) $$f(t)=2^{3 t+4}$$

Step-by-Step Solution

Answer

\(f(t) = 16 \cdot 8^t\)

Step 1: Identify the Exponent Rule

We begin by recognizing the relevant exponent rule: \(a^{x+y} = a^{x} \cdot a^{y}\). This rule helps us separate the exponent into parts that allow for factoring constants and applying the formula.

Step 2: Apply the Exponent Rule

Apply the rule to the given function: \(f(t) = 2^{3t+4}\). Break down the exponent \(3t+4\) into \(3t\) and \(4\), so we can express this as \(2^{3t} \cdot 2^4\).

Step 3: Simplify the Expression

First, recognize \(2^4\) as a constant. Calculating \(2^4\) gives 16. Now, we have: \(f(t) = 2^{3t} \cdot 16\).

Step 4: Rearrange to Standard Form

The function is now in the form \(f(t) = k \cdot a^t\), where \(k=16\) and \(a = 2^3 = 8\). This gives us the final function: \(f(t) = 16 \cdot 8^t\).

Key Concepts

Exponent Rules

Exponents are a shorthand way of expressing repeated multiplication. The core rules around exponents help us handle expressions involving powers more easily. When faced with two powers of the same base added together, the rule we use is \(a^{x+y} = a^x \cdot a^y\). This allows us to split the exponent and simplify expressions.
For example, in the expression \(2^{3t+4}\), the exponent is a sum: \(3t+4\). To split this, you apply the rule to get \(2^{3t} \cdot 2^4\), transforming a single expression into a product of two simpler expressions. This makes further manipulations, like discovering constants or separating variables, feasible.
Understanding the splitting of exponents gives you a handy tool for rewriting complex functions, making it easier to see hidden structures or constants.

Function Transformation

Function transformation involves manipulating functions to present them in a different way. The goal is often to identify specific components or forms that reveal more about the function's behavior. Here, the target was to write the function \(f(t) = 2^{3t+4}\) in the form \(f(t) = k \cdot a^t\), which can reveal growth factors more clearly.
First, you separate the exponent like so: \(2^{3t} \cdot 16\) where \(16\) emerged from calculating \(2^4\). This reveals that \(16\) is a constant factor, and the remaining expression \(2^{3t}\) suggests further transformation.
Recognize that \(2^{3t}\) can itself be seen as \((2^3)^t\). This step changes the base from \(2\) to \(2^3\), simplifying to: \((2^3) = 8\). Thus, the exponential expression is transformed into a function of the form \(f(t) = 16 \cdot 8^t\), completing the transformation process.

Simplification of Expressions

Simplification is all about reducing an expression to its most straightforward form, without altering its value. It makes expressions easier to handle, compute, and understand. In the given exercise, after using the exponent rules, the expression \(2^{3t+4}\) was initially broken down to \(2^{3t} \cdot 16\).
This process reveals a key aspect: calculating constants such as \(2^4\), which equals \(16\).
Then, consider this: \(2^{3t}\) can be restated as \((2^3)^t = 8^t\). This restatement simplifies the expression greatly, offering a form where the base \(8\) is much clearer, and the function is noted as \(f(t) = 16 \cdot 8^t\).
The simplification journey took a complex power and translated it into a manageable expression illustrating the importance of recognizing and manipulating constants and powers within mathematical functions.