Problem 60 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (f \circ g)(-3) $$

Step-by-Step Solution

Answer

16

Step 1: Understand Composition of Functions

The composition of two functions \(f\) and \(g\) is written as \(f \circ g\), which means we apply \(g(x)\) first and then apply \(f(x)\) to the result. In mathematical terms, \( (f \circ g)(x) = f(g(x)) \). Our goal is to find \( (f \circ g)(-3) \).

Step 2: Evaluate g(-3)

First, find \( g(-3) \) using the function \( g(x) = x^2 + x \):\[ g(-3) = (-3)^2 + (-3) \] Calculate each part: \( (-3)^2 = 9 \) and \( 9 + (-3) = 6 \). So, \( g(-3) = 6 \).

Step 3: Evaluate f at the Result from g

Now, plug the result of \( g(-3) = 6 \) into \( f(x) \):\[ f(6) = 3(6) - 2 \]Perform the calculations: \( 3 \times 6 = 18 \) and \( 18 - 2 = 16 \). Thus, \( f(6) = 16 \).

Step 4: Conclusion

Thus, \( (f \circ g)(-3) = 16 \). The final value after computing the composition of the functions at \(-3\) is 16.

Key Concepts

Algebra

Algebra forms the foundational framework for understanding many mathematical concepts, including function composition. It provides a systematic way of handling equations and expressions. In algebra, symbols and letters are used to represent numbers and quantities in formulae and equations. This branch of mathematics enables us to manipulate these symbols to solve problems.

Key areas in algebra involve understanding:

• Expressions: These are combinations of symbols that represent a quantity. For example, \(3x - 2\) in \(f(x) = 3x - 2\).
• Equations: These express a relationship between different expressions. For instance, in \(f(x)=3x-2\), the equals sign represents a balance between the left and right sides.
• Functions: Algebra allows us to describe functions, which are specific types of equations that show how one quantity changes with another.

In this exercise, algebraic manipulation helps us substitute values into functions to find results. By understanding algebraic principles, we can easily follow through the steps to find \((f \circ g)(-3)\) using given functions \(f(x)\) and \(g(x)\).

Evaluate Functions

To evaluate a function means to find the output value of a function for a specified input. In simpler terms, you're determining what the function spits out when you feed it a particular value.

In our exercise, you need to evaluate two functions, \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\), particularly looking at how they work when composed. Evaluating these functions involves:

• Substituting the input into the function. In step 2 of our solution, we plugged \(-3\) into \(g(x)\) to get \(g(-3) = 6\).
• Performing arithmetic operations as dictated by the function's rule. Here, \((-3)^2 + (-3)\) was simplified to \(6\).
• Reusing the output as an input to another function when dealing with compositions. The result from \(g(-3)\) was used as the input to evaluate \(f(6)\).

By following these steps, evaluating functions becomes a straightforward process, crucial for solving composition problems and many algebraic tasks.

Composition of Functions

Function composition is an intriguing and essential concept in algebra. It involves creating a new function by applying one function to the results of another. The composition of functions \(f\) and \(g\) is denoted by \((f \circ g)(x)\), which means "first apply \(g\), then apply \(f\) to the outcome."

This approach simplifies complex operations by breaking them down into manageable parts. Here's what you need to understand:

• The order matters. In \((f \circ g)(x)\), you always apply \(g\) first, then \(f\).
• Each function has its rule, such as \(g(x) = x^2 + x\). You have to apply the function as per its rule to the input.
• The result of applying \(g\) becomes the input for \(f\), simplifying the process of handling multiple operations.

Our exercise shows that by using function composition, you can obtain \((f \circ g)(-3) = 16\) through careful calculation of each function step-by-step. This structured approach allows for understanding complex relationships between variables in algebraic expressions.