Problem 53 - Precalculus Essentials

Question

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

Since the value given, -2, is less than 0, use the first part of the piecewise function. So, \[f(-2)=3(-2)+5=-1\]

Step 2: Evaluate \(f(0)\)

Since the value given, 0, is equal to 0, use the second part of the piecewise function. So, \[f(0)=4(0)+7=7\]

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

Since the value given, 3, is greater than 0, use the second part of the piecewise function. So, \[f(3)=4(3)+7=19\]

Key Concepts

Evaluating Functions

When it comes to understanding functions in mathematics, evaluating them is a fundamental skill. To evaluate a function means to find the value of the function's output based on a specific input. This process involves substituting the given value into the function and performing the arithmetic operations as indicated.

For example, in the case of the piecewise function
\[f(x) = \begin{cases} 3x + 5 & \text{if } x < 0 \ 4x + 7 & \text{if } x \geq 0 \end{cases}\],
to evaluate \( f(-2) \), you replace the \( x \) with -2 in the appropriate part of the function—in this case, the first part because -2 is less than 0, giving us \( f(-2) = 3(-2) + 5 = -1 \). This calculation involves following order of operations, also known as PEMDAS—parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). Remember, evaluating functions is like following a recipe; each step must be done in the correct order to get the intended result.

Function Notation

Function notation is a way of representing functions that emphasizes the input-output relationship. It is a concise method that uses the symbol \( f(x) \) to denote a function named \( f \) with variable \( x \). The letter \( f \) is the function's name, while \( x \) is the placeholder for the input value. When you see \( f(3) \), this is read as 'the value of function \( f \) at 3' and implies calculation should be based on the input of 3.

Function notation is especially helpful with piecewise functions like \[f(x) = \begin{cases} 3x + 5 & \text{if } x < 0 \ 4x + 7 & \text{if } x \geq 0 \end{cases}\],where different rules apply for different intervals of \( x \). For instance, \( f(3) \) requires us to use the second rule due to the input being greater than 0, thus \( f(3) = 4(3) + 7 = 19 \). It is essential to identify which part of the function applies to the input value before carrying out the evaluation.

Domain of a Function

The domain of a function is the set of all possible input values (usually \( x \)) for which the function is defined. In simpler terms, it's all the values you can plug into the function without getting an error, like division by zero or the square root of a negative number in real number calculations.

The domain of the piecewise function \[f(x) = \begin{cases} 3x + 5 & \text{if } x < 0 \ 4x + 7 & \text{if } x \geq 0 \end{cases}\] is all real numbers because both pieces of the function, \( 3x+5 \) and \( 4x+7 \), are defined for all real numbers. However, in other functions, the domain might be limited, for example, by specifying that \( x \) must be greater than or equal to zero. It is crucial to understand the domain as it tells us the scope within which the function operates and ensures we only evaluate it for values within this domain.