Problem 19 - Algebra and Trigonometry

Question

Evaluate the function at the indicated values. $$ \begin{array}{l}{f(x)=2 x+1} \\ {f(1), f(-2), f\left(\frac{1}{2}\right), f(a), f(-a), f(a+b)}\end{array} $$

Step-by-Step Solution

Answer

\(f(1)=3, f(-2)=-3, f\left(\frac{1}{2}\right)=2, f(a)=2a+1, f(-a)=-2a+1, f(a+b)=2a+2b+1\).

Step 1: Understand the Function

The function given is \( f(x) = 2x + 1 \). This is a linear function where \( x \) is the variable. The task is to evaluate this function for various inputs.

Step 2: Evaluate \( f(1) \)

Substitute \( x = 1 \) into the function: \[ f(1) = 2(1) + 1 = 2 + 1 = 3. \] Thus, \( f(1) = 3 \).

Step 3: Evaluate \( f(-2) \)

Substitute \( x = -2 \) into the function: \[ f(-2) = 2(-2) + 1 = -4 + 1 = -3. \] Thus, \( f(-2) = -3 \).

Step 4: Evaluate \( f\left(\frac{1}{2}\right) \)

Substitute \( x = \frac{1}{2} \) into the function: \[ f\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right) + 1 = 1 + 1 = 2. \] Thus, \( f\left(\frac{1}{2}\right) = 2 \).

Step 5: Evaluate \( f(a) \)

Substitute \( x = a \) into the function: \[ f(a) = 2a + 1. \] Thus, \( f(a) = 2a + 1 \).

Step 6: Evaluate \( f(-a) \)

Substitute \( x = -a \) into the function: \[ f(-a) = 2(-a) + 1 = -2a + 1. \] Thus, \( f(-a) = -2a + 1 \).

Step 7: Evaluate \( f(a+b) \)

Substitute \( x = a+b \) into the function: \[ f(a+b) = 2(a+b) + 1 = 2a + 2b + 1. \] Thus, \( f(a+b) = 2a + 2b + 1 \).

Key Concepts

Linear Functions

A linear function is one of the simplest types of mathematical functions. It is represented as a straight line when graphed on a Cartesian coordinate system. Linear functions generally have the form \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept. The function we are dealing with in the problem is \( f(x) = 2x + 1 \), a typical example of a linear function.
Linear functions are characterized by:

• A constant rate of change or slope.
• A straight-line graph that is neither curving nor bending.
• Only one degree variable; that is, the term with the highest power is \( x^1 \).

Recognizing these features helps in understanding why the function behaves consistently over various inputs.

Substitution Method

The substitution method is a straightforward technique frequently used in algebra to replace variables with given values. This method is essential for evaluating functions like the one we're exploring. In the problem, we substitute different values into the function \( f(x) = 2x + 1 \) to find specific function values.
Here's how it typically works:

• Identify the variable in the function you need to replace, like \( x \) in \( f(x) = 2x + 1 \).
• Substitute the specified value for the variable, performing any necessary arithmetic operations.
• Calculate and simplify to find the result of the function with the given input.

By applying the substitution method, we can evaluate the function at any point, providing clarity on the function's value based on the input substituted.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can involve numbers, variables, and operations. In the function \( f(x) = 2x + 1 \), both terms "2x" and "1" form an algebraic expression. When substituting for different values of \( x \), the algebraic expression adjusts accordingly.
Here are some aspects about algebraic expressions that are handy:

• They can be simplified by performing arithmetic on the constants and coefficients.
• Terms in an expression are usually separated by addition (+) or subtraction (−) symbols.
• You can perform operations like addition, subtraction, multiplication, and division with them.

Understanding how algebraic expressions function paves the way for mastering more complex mathematics topics. It also offers insights into how algebraic operations transform expressions during evaluation.

Function Notation

Function notation is a simple yet crucial algebra concept that makes it easier to denote functions in mathematics. It employs symbols to indicate the inputs and corresponding outputs, typically written as \( f(x) \). This notation is handy for expressing that a function \( f \) takes an input \( x \) and yields a particular output.
Some critical points about function notation include:

• The letters \( f, g, \) and \( h \) are frequently used to denote functions, but other letters can be used as well.
• The expression inside the parentheses—such as \( x \) in \( f(x) \)—is the variable or placeholder showing what input the function acts upon.
• Using proper function notation clarifies the function's purpose and the specific variable it uses.

Employing function notation properly ensures precision and consistency when working with various mathematical functions, making it easier to understand and communicate different mathematical operations.