Problem 24 - Technical Mathematics with Calculus

Question

Substitute the given numerical value into each function. $$\text { If } f(x)=15 x+9, \text { find } f(3)$$

Step-by-Step Solution

Answer

\(f(3) = 54\)

Step 1: Identify the Function and Input Value

Identify the function given in the problem, which is \(f(x) = 15x + 9\) and the value to be substituted into the function, which is \(x=3\).

Step 2: Substitute the Input Value into the Function

Substitute \(x=3\) into the function in place of every instance of \(x\). The function becomes \(f(3) = 15(3) + 9\).

Step 3: Perform the Arithmetic Operations

Perform the arithmetic operations to find the result: \(f(3) = 15(3) + 9\) gives \(f(3) = 45 + 9\), which simplifies to \(f(3) = 54\).

Key Concepts

Substituting Values

Substituting values into an algebraic expression is a fundamental skill in mathematics. It involves replacing variables with given numbers to evaluate the expression. For example, consider the function f(x) = 15x + 9. When we are asked to find f(3), we replace every instance of the variable x with the number 3. This process is essential for understanding how functions behave with different inputs and is commonly used to graph functions or solve equations.

When substituting, it is crucial to maintain the integrity of the original expression. If the variable appears more than once, it must be replaced consistently each time. In practice, we often use parentheses to ensure that the substitution is clear and to avoid any mistakes with order of operations.

Arithmetic Operations

Once we have substituted the values, we then perform arithmetic operations to find the result. Arithmetic operations include addition, subtraction, multiplication, and division. In our example, f(3) = 15(3) + 9, we first multiply 15 by 3, then add 9. The correct order of operations (often remembered by the acronym PEMDAS which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) is key to obtaining the right answer. Sometimes, simplifying the expression may involve combining like terms or factoring. Performing these operations accurately is an essential component of evaluating functions and solving algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. These expressions represent values that can change depending on the variables involved. For instance, the expression 15x + 9 is algebraic because it contains the variable x along with numbers and the operations of multiplication and addition.

Algebraic expressions do not have an equality sign as equations do. They are used to define functions and to express mathematical relationships that we can manipulate using algebraic rules. Learning to work with these expressions is crucial for solving more complex problems in mathematics including those in algebra, calculus, and beyond.

Mathematical Functions

A function in mathematics is a special type of relation between sets, where each input (often labeled x) is assigned exactly one output (often labeled f(x)). Functions are often expressed as algebraic expressions, like the example f(x) = 15x + 9. The function describes a rule for how to transform the input into the output.

Understanding how to evaluate functions is critical in mathematics. It involves substituting values, performing arithmetic operations, and interpreting the results. Functions can be used to model real-world situations and can take various forms including linear, quadratic, polynomial, exponential, and more, each with their own unique properties.