Problem 74 - Algebra 2

Question

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (f \circ g)(x) $$

Step-by-Step Solution

Answer

The simplified form of the composition \( (f \circ g)(x) \) equals \( 2x + 28 \).

Step 1: Identify the given functions

We have three functions given: \(f(x)=4 x, g(x)=\frac{1}{2} x+7, h(x)=|-2 x+4|\). However, for this exercise, we only need to focus on functions \( f \) and \( g \), since we need to simplify the composition \( (f \circ g)(x) \).

Step 2: Rewrite the composition of the functions

The composition \( (f \circ g)(x) \) can be rewritten as \( f(g(x)) \). It means replacing every \( x \) in \( f \) with \( g(x) \).

Step 3: Substitute \( g(x) \) into \( f(x) \) to get \( f(g(x)) \)

To substitute \( g(x) \) into \( f(x) \), replace the \( x \) in \( f(x) \) with \( g(x) \). So, instead of \( f(x) = 4x \), we will have \( f(g(x)) = 4(g(x))\).

Step 4: Simplify the function \( f(g(x)) \)

Now we just need to put the function \( g(x) = \frac{1}{2}x + 7 \) into \( 4(g(x)) \). Hence, \( f(g(x)) = 4(\frac{1}{2}x + 7) = 2x + 28 \).

Key Concepts

Linear Functions

Linear functions are fundamental in algebra and calculus, characterized by their straight-line graphs and simple algebraic expressions. They are expressed in the form \( f(x) = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. Understanding linear functions is crucial because they serve as the foundation for more complex mathematical concepts.

In the given exercise, both \( f(x) = 4x \) and \( g(x) = \frac{1}{2}x + 7 \) are linear functions.

• For \( f(x) \), the slope \( m = 4 \) and the intercept \( b = 0 \), meaning it passes through the origin.
• For \( g(x) \), the slope \( m = \frac{1}{2} \) and the intercept \( b = 7 \), indicating it crosses the y-axis at 7.

Linear functions are simple yet powerful, making them an important tool to learn algebraic manipulation such as function composition and simplification.

Function Simplification

Function simplification involves reducing complex expressions to simpler, more manageable forms. This can greatly aid in understanding and solving mathematical problems efficiently.

In the exercise, we simplified the composite function \( (f \circ g)(x) \), meaning we found \( f(g(x)) \).

• The first step was substituting \( g(x) = \frac{1}{2}x + 7 \) into \( f(x) = 4x \), giving \( f(g(x)) = 4(g(x)) \).
• Next, we expanded \( 4(g(x)) \) to arrive at \( 4(\frac{1}{2}x + 7) \).
• Finally, simplifying this expression yields \( 2x + 28 \).

Simplification helps not just in calculating the results, but in providing understanding of the relationships between functions, especially during composition.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra, allowing representation and manipulation of mathematical ideas.

When working with the composition of \( f(x) \) and \( g(x) \) in the exercise, you created a new algebraic expression.

• The initial expressions \( 4x \) and \( \frac{1}{2}x + 7 \) were combined through substitution.
• After substitution, simplification led to the new expression \( 2x + 28 \).
• This new expression represents the output of \( (f \circ g)(x) \) and demonstrates how algebraic manipulation can transform expressions.

Grasping how to form and simplify algebraic expressions is vital for transitioning to more advanced mathematics, enabling deeper insights and solutions for complex problems.