Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answers to these problems are not unique. \(h(x)=3^{2 x}+3^{x}+1\)

Step-by-Step Solution

Answer

The functions are \(f(u) = u^{2} + u + 1\) and \(g(x) = 3^{x}\).

Step 1: Finding Function \(g(x)\)

To start with, we note that if we let \(u = 3^{x}\), it simplifies the expression \(h(x)\) to \(u^{2} + u + 1\).Therefore the function \(g(x)\) can be \(3^{x}\). Hence, \(g(x) = 3^{x}\). This function \(g(x)\) is different from the identity function \(x\).

Step 2: Finding Function \(f(u)\)

Then, with the function \(g(x)\) found, we move onto finding the function \(f(u)\). Looking at the modified expression, \(u^{2} + u + 1\), we can use this as function \(f(u)\) because it takes the output of \(g(x)\) \(i.e. u\) as an input and returns \(u^{2} + u + 1\). So, the function \(f(u)\) that needs to be found is \(f(u) = u^{2} + u + 1\). This function \(f(u)\) also differs from the identity function \(u\).

Step 3: Double Check

Finally, we double check that the \(f(g(x))\) format represents the given function \(h(x)\): Plug \(g(x) = 3^{x}\) into function \(f(g)\) resulting \(f(g(x)) = (3^{x})^{2} + 3^{x} + 1 = 3^{2x} + 3^{x} + 1 = h(x)\), and make sure that neither \(f\) nor \(g\) is an identity function.

Key Concepts

Identity Function

The identity function is a basic concept in mathematics where every element maps to itself. This means that for every input, the output remains unchanged. If the identity function is represented as \( I(x) \), then \( I(x) = x \) for all values of \( x \). This unique property makes it quite intuitive and straightforward.

• The identity function is often used as the basis for understanding more complex functions because it's the simplest function possible.
• In function composition, the identity function can act as a neutral element, meaning when you compose any function with the identity function, you get the original function back.
• It serves as a reminder of the input values without any transformation or alteration.

In the context of the exercise, neither of the functions \( f \) nor \( g \) is an identity function, meaning both must transform the input in some way. This led to the choice of \( g(x) = 3^{x} \) and \( f(u) = u^2 + u + 1 \), both of which are non-trivial functions that do not simply return their input unchanged.

Exponential Function

The exponential function is a powerful and widely-used function in mathematics. It is characterized by its base raised to the power of the input variable. With the general form \( a^x \), where \( a \) is a constant and \( x \) is the exponent, it describes a rapidly increasing or decreasing curve, depending on the base.

• One of the most common exponential functions involves the base number \( e \), called the natural exponential function \( e^x \), but any constant can be used as the base, such as \( 3 \, ext{in this exercise.} \)
• Exponential functions are involved in many real-life phenomena, such as population growth, radioactive decay, and compound interest.
• These functions grow much faster than polynomial functions and are crucial in understanding rates of change.

In the exercise, the function \( g(x) = 3^{x} \) is an example of an exponential function. It takes the input \( x \) and raises 3 to the power of \( x \), transforming it significantly, which clearly differentiates it from an identity function.

Quadratic Function

The quadratic function is a key concept in algebra and calculus with its general form being \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Its graph is a parabola, which can open upwards or downwards depending on the sign of \( a \).

• When \( a > 0 \), the parabola opens upwards, resembling a U-shape, and when \( a < 0 \), it opens downwards.
• Quadratic functions are used to model various situations, such as projectile motion, area calculations, and optimization problems.
• They can always be expressed in terms of their vertex form or factored form, which are useful for solving specific types of quadratic equations.

In the context of the exercise, the function \( f(u) = u^2 + u + 1 \) represents a quadratic function in the variable \( u \), showing interaction between linear and squared terms. This function takes the output of the exponential \( g(x) \) function and further transforms it, ultimately leading to the composition that matches \( h(x) \).