Problem 30 - College Algebra with Modeling and Visualization

Question

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. $$ f(0)=7 \text { and } f(-1)=1 $$

Step-by-Step Solution

Answer

The function is \(f(x) = 7 \cdot 7^x\).

Step 1: Use the first condition

We know from the condition that \(f(0) = 7\). Substituting \(x = 0\) in the equation \(f(x) = C \cdot a^x\), we get:\[ C \cdot a^0 = C \cdot 1 = C = 7 \]Hence, \(C = 7\).

Step 2: Use the second condition

Now, using the second condition \(f(-1) = 1\), we substitute into the equation \(f(x) = 7 \cdot a^x\) since we found \(C = 7\):\[ 7 \cdot a^{-1} = 1 \]Simplifying gives:\[ \frac{7}{a} = 1 \]

Step 3: Solve for \(a\)

To find \(a\), solve the equation from Step 2:\[ \frac{7}{a} = 1 \]Multiplying both sides by \(a\), we have:\[ 7 = a \]Therefore, \(a = 7\).

Step 4: Write the function \(f(x)\)

Substitute \(C = 7\) and \(a = 7\) back into the original equation, we find the function:\[ f(x) = 7 \cdot 7^x \]

Key Concepts

Exponentiation

Exponentiation is the mathematical operation that involves raising a number, known as the base, to the power of an exponent. When exploring exponential functions, such as the one in our exercise, understanding how exponentiation works is key.

In the function form \[f(x) = Ca^x\], \(a^x\) represents an exponential operation where \(a\) is the base and \(x\) is the exponent. Here’s a quick breakdown:

• When \(x = 0\), the result \(a^0\) aleays equals 1, regardless of the value of \(a\), as long as \(a\) is not zero.
• When \(x\) is negative, such as \(-1\), the result \(a^{-1}\) is the reciprocal of the base, \(\frac{1}{a}\).

These principles allow us to determine values like \(C\) and \(a\) from given function conditions, as seen when solving the exercise step by step.

Equation Solving

Equation solving involves finding the unknown variable that satisfies a mathematical equation. In our exercise, we were tasked with finding the constants \(C\) and \(a\) of the function \[f(x) = C a^{x}\].

The process usually involves:

• Substituting known values into the equation to simplify and solve it.
• Step 1 revealed \(C\) as 7 by using \(f(0) = 7\). Since \(a^0 = 1\), substituting \(x = 0\) led to \(C = 7\).
• In Step 2, we used the second condition \(f(-1) = 1\) to find \(a\). By substituting \(x = -1\), you determine that the equation \(7a^{-1} = 1\) simplifies to \(a = 7\).

This structured approach helped solve for the unknowns effectively by isolating each variable step-by-step.

Function Properties

An understanding of function properties enables us to predict and comprehend the nature of functions. For exponential functions of the form \[f(x) = C a^x\], several properties are important to understand:

• The base \(a\) determines the growth or decay nature of the function.
• For \(a > 1\), the function describes exponential growth, seen here where \(a = 7\).
• An exponential function is continuous and smooth.
• These functions typically have a horizontal asymptote. In this exercise, notice that as \(x\) tends toward negative infinity, \(f(x)\) approaches zero, but never actually reaches it, aligning with that property.

Recognizing these characteristics allows better modeling and understanding of natural phenomena, financial growth, or decay processes that are well represented by exponential expressions.