Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f g)(-2)\)

Step-by-Step Solution

Answer

The solution to \((f g)(-2)\) is 5.

Step 1: Evaluate g(-2)

In this step, we will substitute -2 into the function \(g(x)=x^{2}-2\), so \(g(-2)=(-2)^{2}-2=4-2=2.\)

Step 2: Substitute g(-2) into f

Now we take the result of \(g(-2)\), which is 2, and substitute it into function \(f\). So, \(f(g(-2))=f(2)=2(2)+1=5.\)

Key Concepts

Composite Functions

In mathematics, composite functions are a way of combining two functions into a single expression. Imagine having two functions, say \( f(x) \) and \( g(x) \). A composite function is formed by applying one function to the results of another. It is denoted as \( (f \circ g)(x) \), which means \( f(g(x)) \). This concept allows you to "stack" functions on top of each other, applying them in sequence.
The order in which you apply these functions is important and can change the outcome. For example, in our original exercise, you first find \( g(-2) \) and then apply the result to \( f \). This relates to the idea of function composition because you end up evaluating \( f(g(-2)) \).
Understanding how composite functions work is crucial in different branches of mathematics, including calculus and advanced algebra. They help in understanding complex function manipulations and make solving elaborate mathematical problems more intuitive.

Substitution Method

The substitution method is a systematic approach to solve composite functions. It involves replacing a variable with a particular value or another expression to simplify the problem. In our exercise, substitution helps in evaluating \( fg(-2) \).
Here's how the substitution method works step by step:

• First, substitute the value into the function \( g(x) \) to find \( g(-2) \).
• After calculating \( g(-2) \), you take this result and substitute it into the function \( f(x) \) to find \( f(g(x)) \).

The method can simplify complicated functions by breaking them down into more manageable parts. With practice, substitution becomes a handy tool when handling nested functions or solving equations involving complex expressions.

Algebraic Functions

Algebraic functions, like \( f(x) = 2x+1 \) and \( g(x) = x^2-2 \) in our problem, are defined by algebraic expressions involving variables and constants. They are the building blocks of most algebra problems you'll encounter.
These functions can be as simple as linear (one degree, like \( f(x) = 2x + 1 \)) or more complex, like quadratic functions (degree two, as in \( g(x) = x^2 - 2 \)). Linear functions represent straight lines, while quadratic functions form parabolas when graphed. Both appear frequently in mathematics and various applications.
Evaluating algebraic functions involves substituting specific values into the function equation and solving it. Understanding how to manipulate these functions using basic algebraic rules is crucial. This forms the foundation for solving real-life problems and understanding more complex functions.