Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(\mathrm{f} : x \rightarrow|3 x-7|\)

Step-by-Step Solution

Answer

The expression for \( f(x) \) is \( |3x - 7| \), and \( f(5) = 8 \).

Step 1: Identify the function

The function provided is given by the expression \( f(x) = |3x - 7| \). It describes the absolute value of the linear expression \( 3x - 7 \).

Step 2: Evaluating the function at x = 5

To find \( f(5) \), substitute \( x = 5 \) into the function \( f(x) = |3x - 7| \). This gives us \( f(5) = |3(5) - 7| \).

Step 3: Simplify inside the absolute value

Calculate the expression inside the absolute value: \( 3(5) = 15 \), so \( 3(5) - 7 = 15 - 7 \), which simplifies to \( 8 \).

Step 4: Apply the absolute value

Since \( 8 \) is already a positive number, the absolute value does not change it. Therefore, \( |8| = 8 \).

Step 5: Conclusion for \( f(5) \)

We find that \( f(5) = 8 \). This is the value of the function when \( x = 5 \).

Key Concepts

Evaluating Functions

When we talk about evaluating functions, we're discussing how to find the output of a function for a specific input. This concept is crucial in understanding how functions work. It involves substituting a given value of the variable into the function expression. For example, in our exercise, the function is given by \( f(x) = |3x - 7| \).

To evaluate \( f(5) \), substitute \( x = 5 \) into the function. So, you'd calculate: \( f(5) = |3(5) - 7| \).

• First, calculate the expression inside the absolute value: \( 3(5) = 15 \), so \( 15 - 7 = 8 \).
• Next, apply the absolute value, which stays the same since \( 8 \) is positive: \( |8| = 8 \).

Therefore, the evaluated result of the function for \( x = 5 \) is \( 8 \). Evaluating functions is all about inputting values and calculating the resulting output accurately.

Linear Expressions

Linear expressions are algebraic expressions where the variable is exposed to the first power, meaning they have no exponents other than one and they do not involve any complex functions like squares or square roots. These expressions typically form straight lines when graphed on a coordinate plane. In our function \( f(x) = |3x - 7| \), the piece inside the absolute value, \( 3x - 7 \), is a linear expression.

Characteristics of Linear Expressions:

• The most basic form of a linear expression is \( ax + b \), where \( a \) and \( b \) are constants.
• In this example, \( 3x - 7 \), \( 3 \) is the coefficient of the variable \( x \), and \(-7\) is the constant term.
• Linear expressions are easy to manipulate, solve, and graph because of their simplicity.

The linear part of the absolute value functions, like \( 3x - 7 \), makes understanding and solving them straightforward. Understanding linear expressions is key to working with more complex function types like polynomial and absolute value functions.

Function Expressions

Function expressions define relationships between inputs and outputs. They help describe how one set of values, the input, is related to another set, the output. A function is a rule that assigns each input exactly one output. For the given function, \( f(x) = |3x - 7| \), the expression tells us how to operate on any given \( x \) to find \( f(x) \).

Understanding Function Expressions:

• The notation \( f(x) \) is read as "f of x," meaning the function of variable \( x \).
• The vertical bar symbols \( | | \) indicate that we are dealing with an absolute value, which always results in a non-negative outcome.
• Function expressions can represent various types of mathematical relationships, including linear equations, quadratics, polynomials, and can even incorporate trigonometry.

Comprehension of function expressions is fundamental for anyone studying algebra. It enables predictions about behavior, like trajectory or trend, and acts as a building block for more advanced mathematical concepts. Recognizing and working with function expressions is essential for solving a wide array of math problems efficiently.