Problem 61 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Evaluate the function for the given value of \(x\). $$f(x)=x^3-2 x+7 ; x=-2$$

Step-by-Step Solution

Answer

The value of the function at \(x = -2\) is 3.

Step 1: Substitution

Substitute \(x = -2\) in function \(f(x) = x^3 - 2x + 7\). This result in \(f(-2) = (-2)^3 - 2(-2) + 7\).

Step 2: Evaluation

Evaluate the right side of the equation. Calculate the cube of -2, multiply -2 by -2, and add all results together. This gives \(f(-2) = -8 + 4 + 7\).

Step 3: Final Calculation

Lastly, calculate the result of the right side of the equation, \(f(-2) = -8 + 4 + 7\), to get \(f(-2) = 3\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra used to evaluate functions. It involves replacing the variable in a function with a given number to compute the function's output. For example, if asked to evaluate the function f(x) = x^3 - 2x + 7 for x = -2, you apply the substitution method by replacing every instance of x in the function with -2.

The process is straightforward: take the given function and wherever you see the variable x, put the value -2 in its place, taking care to follow the order of operations – parentheses, exponents, multiplication and division, and then addition and subtraction (PEMDAS). Once all instances of x are substituted, you can evaluate the arithmetic to find your answer.

Function Notation

Function notation is a way to denote functions in a concise and standardized form, typically using the letter f, followed by parentheses containing the variable, such as in f(x). This notation is helpful because it clearly specifies which variable is the input, and it indicates that f is a function related to x. When using function notation, the formula within the parentheses describes the relationship between the input and the output.

For instance, in the function f(x) = x^3 - 2x + 7, f(x) denotes the function name and formula, and x is the input variable. If we want to find the function's value for x=-2, we write f(-2), signifying that -2 is the input to function f.

Polynomial Functions

Polynomial functions are algebraic expressions that include terms with variables raised to whole number exponents, and possible constant terms. For example, the function f(x) = x^3 - 2x + 7 is a polynomial function because it consists of x raised to the third power, x to the first power (implicitly as x), and a constant, 7.

Polynomials are classified by their degree, which is the highest exponent of the variable in the function. In this case, the degree is three because the highest exponent on x is 3. Polynomial functions of degree three or higher can have complex curves and turns when graphed, and they can cross the x-axis more than once, indicating they may have multiple real roots. By understanding the structure of polynomial functions, students can better analyze and graph these types of equations.