Problem 7 - Applied Calculus

Question

For the functions \(f\) and \(g\) in Problems \(7-10,\) find (a) \(\quad f(g(1))\) (b) \(g(f(1))\) (c) \(\quad f(g(x))\) (d) \(g(f(x))\) (e) \(f(t) g(t)\) $$f(x)=x^{2}, g(x)=x+1$$

Step-by-Step Solution

Answer

(a) 4, (b) 2, (c) \(x^2 + 2x + 1\), (d) \(x^2 + 1\), (e) \(t^3 + t^2\)

Step 1: Evaluate f(g(1))

To find \(f(g(1))\), start by calculating \(g(1)\). The function \(g(x) = x + 1\), so \(g(1) = 1 + 1 = 2\). Next, evaluate \(f(2)\) using \(f(x) = x^2\). Thus, \(f(2) = 2^2 = 4\).

Step 2: Evaluate g(f(1))

First, find \(f(1)\) using \(f(x) = x^2\). Therefore, \(f(1) = 1^2 = 1\). Then substitute into \(g\), where \(g(x) = x + 1\). Thus, \(g(f(1)) = g(1) = 1 + 1 = 2\).

Step 3: Express f(g(x))

To express \(f(g(x))\), substitute \(g(x)\) into \(f\). Since \(g(x) = x + 1\), we need \(f(x + 1)\). So, \(f(x + 1) = (x + 1)^2 = x^2 + 2x + 1\).

Step 4: Express g(f(x))

Substitute \(f(x)\) into \(g\) to find \(g(f(x))\). Since \(f(x) = x^2\), we substitute into \(g\), giving us \(g(x^2) = x^2 + 1\).

Step 5: Evaluate f(t)g(t)

Find the product \(f(t)g(t)\). Using \(f(t) = t^2\) and \(g(t) = t + 1\), the product is \(f(t)g(t) = t^2(t + 1) = t^3 + t^2\).

Key Concepts

Composition of Functions

In mathematics, the composition of functions is a fundamental concept that combines two functions to create a new one. Imagine you have two functions: one called \( f(x) \) and the other \( g(x) \). When these functions are composed, you plug the output of one function into the input of the other. This can be represented as \( f(g(x)) \) or \( g(f(x)) \).

This method helps in simplifying complex expressions and is extensively used to transform equations in various fields, particularly calculus. When composing functions, the order is crucial because \( f(g(x)) \) is generally different from \( g(f(x)) \). In our example, if \( f(x) = x^2 \) and \( g(x) = x + 1 \), then:

• \( f(g(x)) = f(x+1) = (x+1)^2 \)
• \( g(f(x)) = g(x^2) = x^2 + 1 \)

Notice how different these outputs can be simply by changing the order.

Functions Evaluation

Function evaluation is the process of determining the output of a function for a specific input. This is a critical step in function composition or in solving any calculus-related problem. Let's take \( f(x) = x^2 \) and \( g(x) = x+1 \) from our problem.

When you evaluate \( f(g(1)) \), you first compute \( g(1) \). Since \( g(x) = x + 1 \), we substitute 1 for \( x \) yielding \( g(1) = 2 \). Next, input 2 into \( f(x) \). This results in \( f(2) = 2^2 = 4 \).

For \( g(f(1)) \), you start with \( f(1) = 1 \). Plug this into \( g \) to get \( g(1) = 2 \).

• \( f(g(1)) = f(2) = 4 \)
• \( g(f(1)) = g(1) = 2 \)

Evaluating these correctly is key to understanding how composite functions work.

Product of Functions

The product of functions involves multiplying two functions to create a new function. Given two functions \( f(x) \) and \( g(x) \), their product is denoted as \( f(x) \cdot g(x) \).

Using the problem example, if \( f(x) = x^2 \) and \( g(x) = x + 1 \), the product \( f(t)g(t) \) is obtained by multiplying \( t^2 \) and \( t + 1 \). This results in the expression \( t^2(t + 1) = t^3 + t^2 \).

This expression tells us how the resulting function behaves across various values of \( t \), providing a more comprehensive view of the interaction between \( f \) and \( g \). Products of functions are useful in various branches of mathematics to model situations where two independent functions contribute simultaneously.

Calculus Problem-Solving

When working with calculus problems, it's crucial to develop a step-by-step approach to finding solutions. A problem involving function composition, evaluation, and products provides a rich ground for practicing essential calculus skills.

Understanding each step is critical:

• First, determine how to compose functions within given problem constraints.
• Second, evaluate using specific numerical values to verify results.
• Third, apply these operations to more complex expressions involving products.

Breaking down the tasks helps in simplifying and solving a wide range of calculus problems. Consider our example where precise calculations at each stage—such as correctly evaluating \( f(g(1)) \) and determining \( f(t)g(t) \)—illustrate the analytical skills demanded in calculus. By practicing these steps, you build efficiency and confidence in tackling more challenging problems.