Problem 28 - College Algebra with Modeling and Visualization

Question

Find \(C\) and a so that \(f(x)=C a^{x}\) satisfies the given conditions. \(f(1)=3\) and for each unit increase in \(x,\) the output is multiplied by \(\frac{3}{4}\)

Step-by-Step Solution

Answer

\( C = 4 \) and \( a = \frac{3}{4} \).

Step 1: Understand the Function

We are given a function of the form \( f(x) = C \cdot a^x \). This function includes two parameters: \( C \) and \( a \), which are constants that we need to find.

Step 2: Apply the First Condition

We know that \( f(1) = 3 \). Substituting 1 for \( x \) in the function gives us the equation \( C \cdot a^1 = 3 \), or equivalently, \( C \cdot a = 3 \). This is our first equation.

Step 3: Use the Multiplication Factor

For each unit increase in \( x \), the output is multiplied by \( \frac{3}{4} \). This means if \( x \) increases by 1, \( f(x+1) = \frac{3}{4} f(x) \). Using our function, substitute \( x+1 \) in place of \( x \) and set it as \( acf(x) = \frac{3}{4}f(x) \), leading to \( a = \frac{3}{4} \).

Step 4: Find Constant C

Now that we know \( a = \frac{3}{4} \), substitute this value into the equation from Step 2: \( C \cdot \frac{3}{4} = 3 \). Solving for \( C \), we multiply both sides by \( \frac{4}{3} \), obtaining \( C = 3 \times \frac{4}{3} = 4 \).

Step 5: Conclusion: Write Final Answer

We have found that \( C = 4 \) and \( a = \frac{3}{4} \). Thus, the function that satisfies all conditions is \( f(x) = 4 \cdot \left( \frac{3}{4} \right)^x \).

Key Concepts

Function Composition

Function composition is an important concept, especially when dealing with exponential functions like the one in the problem. It involves combining two or more functions to create a new one. This can be visualized as feeding the output of one function into the input of another.
In the exercise, although we don't have a direct composition of multiple functions, the idea of composing factors like constants and variables helps in understanding the changes in the function's behavior when parameters alter.
For instance:

• If we had another function say, \( g(x) \), and we composed it with \( f(x) \), we might represent this as \( f(g(x)) \).
• This means every input \( x \) is first processed by \( g \), and then the output of \( g \) becomes the input for \( f \).

This is useful for chaining processes in a mathematical setting and is especially relevant when dealing with sequences of operations that dynamically affect the growth or decay within a problem.

Algebraic Equations

Solving algebraic equations requires isolating the unknowns by using a series of mathematical operations. In this exercise, you have two unknowns \( C \) and \( a \), which are constants in the exponential function.
Here's how you can approach such problems:

• Identify what each part of the equation represents and what you're solving for.
• Substitute known values into the equation to simplify and create new equations.
• Use manipulation skills like addition, subtraction, multiplication, and division to isolate the unknown variable.
• Check the solution by plugging the values back into the original equation to ensure they satisfy all conditions.

In this case, we first found an equation from the condition \( f(1) = 3 \), leading to \( C \times a = 3 \). Next, we used the information about the function's output changing by a factor of \( \frac{3}{4} \) for every increment of \( x \), aiding us in finding \( a \). Finally, by solving the revised equations, the constants \( C \) and \( a \) were obtained.

Problem Solving

Problem solving in mathematics often involves breaking down a complex problem into more manageable parts. For functions like the one given, this involves understanding the role each component plays and how they interact.
In the given exercise, a systematic approach was used:

• Begin by understanding the problem, including all given conditions and what is being asked.
• Identify relationships between given quantities, such as how the function changes with \( x \).
• Translate these relationships into mathematical equations.
• Systematically solve these equations to find the unknowns using logical reasoning and mathematical operations.

A good practice in problem solving is to visualize the problem or form a diagram if applicable. Conceptualizing the problem in different ways can provide deeper insights and more intuitive solutions. Consistency in solving similar problems with these strategies enhances not only skills but also confidence in tackling complex problems.