Problem 7 - Beginning and Intermediate Algebra

Question

For each pair of functions, find a) \((f g)(x)\) and \(b\) ) \((f g)(-3)\). $$f(x)=2 x+3, g(x)=3 x+1$$

Step-by-Step Solution

Answer

a) The composition of functions is \((fg)(x) = 6x + 5\). b) The value of \((fg)(-3)\) is \(-13\).

Step 1: Find the composition of functions (fg)(x)

To find \((fg)(x)\), we need to substitute the entire \(g(x)\) function into the \(f(x)\) function, like this: \[(fg)(x) = f(g(x))\] Now, substitute the given functions: \[(fg)(x) = f(3x + 1)\] Replace \(x\) in \(f(x)\) with the expression for \(g(x)\): \[(fg)(x) = 2(3x+1) + 3\] Now, simplify the expression: \[(fg)(x) = 6x + 2 + 3\] \[(fg)(x) = 6x + 5\] So, the composition of functions is: \((fg)(x) = 6x + 5\)

Step 2: Find the value of (fg)(-3)

Now that we have the composition function \((fg)(x) = 6x + 5\), we can find the value of \((fg)(-3)\). Substitute \(x = -3\) into the expression: \[(fg)(-3) = 6(-3) + 5\] Now, simplify the expression: \[(fg)(-3) = -18 + 5\] \[(fg)(-3) = -13\] So, the value of \((fg)(-3)\) is \(-13\).

Key Concepts

Algebraic Functions

Algebraic functions are a category of functions composed of polynomials and other algebraic expressions. These functions can be expressed using basic operations like addition, subtraction, multiplication, and division.
For instance, consider the functions given in the exercise:

• \( f(x) = 2x + 3 \)
• \( g(x) = 3x + 1 \)

Both functions are linear and are expressed in the form of \(mx + c\), which is a simple form of an algebraic function. Their primary characteristic is that the highest power of the variable \(x\) is 1, hence categorizing them as linear functions. Understanding these simple algebraic functions makes deciphering more complex compositions easier.

Composition of Functions

The concept of the composition of functions involves creating a new function by using one function as the input for another. It is represented as \((fg)(x)\), meaning function \(f\) acting on the result of function \(g\).
Here’s how you achieve it:

• Substitute \(g(x)\) into \(f(x)\).
• The operation is symbolized as \[ (fg)(x) = f(g(x)) \].

In the provided problem, we did the following steps:

• Start with \(g(x) = 3x + 1\).
• Plug \(g(x)\) into \(f(x) = 2x + 3\), replacing \(x\) with \(3x + 1\).

After substitution and simplification, we arrive at \((fg)(x) = 6x + 5\). This process shows how the interaction of two functions can form a new algebraic expression.

Evaluating Functions

Evaluating functions involves determining the output of a function given a specific input. Once we have the composed function \((fg)(x) = 6x + 5\), we can compute specific values, such as \((fg)(-3)\). Naturally,

• Substitute \(-3\) for \(x\) in \((fg)(x)\).
• Follow through with the arithmetic operations in the equation.

This process of evaluation leads to:

• \(6(-3) + 5 = -18 + 5 = -13\).

Thus, the value of \((fg)(-3)\) is \(-13\). Evaluating functions helps in verifying their behavior for specific inputs and is a routine part of dealing with mathematical expressions.