Problem 22 - Algebra 2 and Trigonometry

Question

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=-x^{3}+x-24 \text { and } a=-3 $$

Step-by-Step Solution

Answer

\(a = -3\) is a root because \(\mathrm{f}(-3) = 0\).

Step 1: Substitute a into the function

To find \(\mathrm{f}(a)\), substitute \(a = -3\) into the function \(\mathrm{f}(x) = -x^3 + x - 24\). This gives:\[ \mathrm{f}(-3) = -(-3)^3 + (-3) - 24. \]

Step 2: Simplify the expression

Calculate \(-(-3)^3\), which is equal to \(-(-27)\) or \(27\). Now, add this to \(-3\) and \(-24\):\[ \mathrm{f}(-3) = 27 - 3 - 24. \]

Step 3: Calculate the total

Perform the addition and subtraction:\[ 27 - 3 = 24, \]and then,\[ 24 - 24 = 0. \]Thus, \(\mathrm{f}(-3) = 0.\)

Step 4: Determine if a is a root

Since \(\mathrm{f}(-3) = 0\), this means that \(a = -3\) is a root of the function \(\mathrm{f}(x) = -x^3 + x - 24\).

Key Concepts

Roots of Functions

Understanding the roots of a function is essential in analyzing polynomial functions. A *root*, often referred to as a *zero* of the function, is any value of the variable that makes the function equal to zero. In simpler terms, for a function \( \mathrm{f}(x) \), a root is such that \( \mathrm{f}(a) = 0 \).

• Roots indicate where the graph of the function intersects the x-axis.
• For polynomial functions, the number of possible roots is determined by the degree of the polynomial.

Finding a root involves evaluating the function for specific values. If the result is zero, then that value is a root. Recognizing roots helps in sketching graph behavior and solving polynomial equations.

Function Evaluation

Function evaluation is the process of determining the output of a function for particular input values. This is a key aspect when working with any algebraic function.

• To evaluate a function, simply replace the variable in the function with the given number.
• Perform the arithmetic operations following the standard order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

For example, substituting \( a = -3 \) into \( \mathrm{f}(x) = -x^3 + x - 24 \) means calculating \( -(-3)^3 + (-3) - 24 \). This involves simplifying each part to find the result, which then aids in determining whether this value is a root of the function.

Cubic Functions

Cubic functions are polynomial functions of the form \( \mathrm{f}(x) = ax^3 + bx^2 + cx + d \), where \( a eq 0 \). These functions are characterized by their degree of three, giving them unique properties.

• Cubic functions can have one to three real roots, indicating where they intersect the x-axis.
• The graph of a cubic function typically features one or two turning points.
• Their shape can vary greatly, stretching or contracting depending on the leading coefficient \( a \).

In the given exercise, the function \( \mathrm{f}(x) = -x^3 + x - 24 \) is a cubic polynomial. Its properties can be analyzed by looking for its roots, evaluating it over different intervals, and using calculus for detailed behavior insights. Understanding these functions equips one to tackle a wide range of algebraic problems effectively.