Problem 37 - College Algebra Essentials

Question

Evaluate each piece wise function at the given values of the independent variable. \(f(x)=\left\\{\begin{array}{ll}3 x+5 & \text { if } \quad x<0 \\ 4 x+7 & \text { if } \quad x \geq 0\end{array}\right.\) a. \(f(-2)\) b. \(f(0)\) c. \(f(3)\)

Step-by-Step Solution

Answer

a. \(f(-2)=-1\), b. \(f(0)=7\), c. \(f(3)=19\)

Step 1: Determine the Appropriate Function for Each Value

Based on the definition of the piecewise function, use the function \(3x+5\) for \(x<0\), and \(4x+7\) for \(x \geq 0\). So, for \(f(-2)\), since -2 is less than 0, use the function \(3x+5\). For \(f(0)\) and \(f(3)\), since 0 and 3 are greater than or equal to 0, use the function \(4x+7\).

Step 2: Substitute the Values for x in the Appropriate Function

Substitute -2 into \(3x+5\) to get \(f(-2)=3(-2)+5=-1\). Substitute 0 and 3 into \(4x+7\) to get \(f(0)=4(0)+7=7\) and \(f(3)=4(3)+7=19\).

Step 3: Solution for Each Part

Based on the substitution in step 2, the solutions for each part are: a. \(f(-2)=-1\), b. \(f(0)=7\), and c. \(f(3)=19\)

Key Concepts

Evaluating Functions

Evaluating functions involves finding the output of a function for specific inputs. In the context of piecewise functions, this means choosing the right formula depending on the given value of the independent variable. Let's break this down simply:

• If you have a function like the piecewise example provided, it’s essential to understand at which part the input value falls.
• When the input for the independent variable aligns with a certain condition, you apply that specific function rule to evaluate the result.

For example, if you’re asked to evaluate \( f(x) \) at an input of \(-2\), and the rule is \( 3x+5 \) if \( x < 0 \), you simply substitute \( -2 \) into \( 3x+5 \) to find the output, which is \(-1\). Basically, selecting the right branch of the function is key to correct evaluation.

Function Notation

Function notation is a way to represent functions in a clear and concise manner. It uses symbols to describe the function rule and input-output relationship, usually noted as \( f(x) \). Here's a closer look:

• The "\( f \)" is the name of the function. It indicates the particular function you are working with.
• The "\( (x) \)" refers to the input that the function considers. This is the independent variable you're substituting numbers into.

Function notation helps to organize complex calculations and clarifies which variable is being used. In our example, \( f(-2) \), the "\( f \)" signifies the function and "\( -2 \)" is the specific input we're using.
Moreover, when used correctly, function notation ensures that when you see an expression like \( f(3) \), you know exactly which function formula is being applied for that value.

Independent Variable

The independent variable in a function is the "input" variable, typically represented by "\( x \)" in mathematical notation. It determines the output when placed within the function.

• In the case of the piecewise function from the exercise, the independent variable \( x \) can take various values, which in turn influence which part of the function is used.
• "Independent" means this variable is the one you can change freely, and each value you select will affect the result of the function.

Understanding the role of the independent variable is crucial when evaluating functions because it directly impacts which rules apply to your situation.
If you consider \( f(0) \) and \( f(3) \), where both use the rule \( 4x+7 \) as \( x \) is \( \geq 0 \), substituting \( 0 \) and \( 3 \) demonstrates how different values of the independent variable yield different outputs.

Algebraic Expressions

Algebraic expressions form the basis of functions by defining the relationship between inputs and outputs. A piecewise function like the one in the problem uses different algebraic expressions depending on the condition of the input.

• For instance, \( 3x+5 \) and \( 4x+7 \) are algebraic expressions showing different pathways a function could take.
• These expressions use operations (addition, multiplication) and numbers to transform the input (independent variable \( x \)) into an output.

Grasping algebraic expressions is crucial because they define how functions behave for different input values. For example, when \( x = -2 \), using \( 3x+5 \) involves performing calculations such as multiplication and addition.
Observing how \( f(x) \) changes when we input different values helps understand the function's overall structure and the impact of selecting one algebraic expression over another.