Problem 6 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The composite function \((f g)(x) = -6x - 2\) and the value of the composite function at -3 is \((f g)(-3) = 16\).

Step 1: Find the composite function (fg)(x)

To find the composite function, we need to find the function f(g(x)). We have: \(f(x) = -2x\) and \(g(x) = 3x + 1\). So, \(f(g(x)) = -2(3x + 1)\).

Step 2: Simplify the composite function (fg)(x)

Now, we will simplify the expression for \(f(g(x))\): \(f(g(x)) = -2(3x + 1)\) \[\Rightarrow f(g(x)) = -6x - 2\]

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

Now we will find the value of the composite function when the input is -3, that is, \(f(g(-3))\): \[f(g(-3)) = -6(-3) - 2\] \[\Rightarrow f(g(-3)) = 18 - 2\] \[\Rightarrow f(g(-3)) = 16\] The final solutions are: a) The composite function: \((f g)(x) = -6x - 2\) b) The value of the composite function at -3: \((f g)(-3) = 16\)

Key Concepts

Function Operations

In mathematics, function operations allow us to combine two or more functions to create new ones. This is fundamental for solving more complex problems. When we refer to the operation of functions, we often talk about:

• Addition and Subtraction: Combining functions by adding or subtracting their outputs.
• Multiplication and Division: Forming a product or quotient of two functions.
• Composition: A key type of operation where one function is applied to the result of another, such as in \( (f \circ g)(x) \) which becomes \( f(g(x)) \).

Here, the focus is on composition, where we compose two functions \( f(x) = -2x \) and \( g(x) = 3x + 1 \). This involves substituting \( g(x) \) into \( f \), illustrating how functions can be layered for more dynamic outcomes.
Practicing these operations helps build a solid foundation for handling more intricate equations in calculus and beyond.

Evaluation of Functions

Function evaluation is the process of finding the output of a function for a given input value. It is a critical step in understanding how functions behave. To evaluate functions effectively, consider:

• Identify the Function: Determine the formula you need, such as \( f(x) = -2x \) or \( g(x) = 3x + 1 \).
• Substitute the Input: Simply replace the input variable with the given number.
• Calculate the Result: Perform the arithmetic to find the answer.

For instance, to find \( f(g(-3)) \):
1. Substitute \( -3 \) into \( g(x) \), giving \( g(-3) = 3(-3) + 1 = -9 + 1 = -8 \).
2. Use this result as the input for \( f(x) \): \( f(-8) = -2(-8) = 16 \).
This approach is step-by-step arithmetic guidance, leading us to the value of the composite function \( f(g(-3)) = 16 \).

Algebraic Functions

Algebraic functions involve algebraic expressions, including variables, constants, and arithmetic operations. They are formulas that produce numbers when given input values. Types of algebraic functions include linear, quadratic, and polynomial functions.

• Linear Functions: These have the form \( ax + b \), reflecting a straight-line graph.
• Quadratic Functions: Characterized by \( ax^2 + bx + c \), creating a parabolic curve.
• Composite Functions: Result from combining two functions, such as \( f(g(x)) \).

In the given exercise, \( f(x) = -2x \) is a linear function, and \( g(x) = 3x + 1 \) is also linear. The composition \( f(g(x)) = -6x - 2 \) remains linear, illustrating how operations affect the form of resulting functions.
By practicing with algebraic functions, students gain insight into how mathematical structures build upon each other, a skill useful in various advanced math topics.