Problem 19 - 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? $$ \mathrm{f}(x)=x^{4}-1 \text { and } a=1 $$

Step-by-Step Solution

Answer

\( a = 1 \) is a root because \( f(1) = 0 \).

Step 1: Substitute 'a' into the function

We have the function \( f(x) = x^4 - 1 \). To find \( f(a) \), where \( a = 1 \), substitute \( a \) for \( x \) in the function: \( f(1) = 1^4 - 1 \).

Step 2: Simplify the expression

Simplify the expression obtained by substituting \( a \) into the function. \( f(1) = 1^4 - 1 = 1 - 1 = 0 \).

Step 3: Determine if 'a' is a root

A number \( a \) is a root of the function if \( f(a) = 0 \). Since \( f(1) = 0 \), this means 1 is a root of the function \( f(x) = x^4 - 1 \).

Key Concepts

Roots of a Function

The roots of a function, also known as zeros or solutions, are the values of the variable that satisfy the equation when the function's output is zero. Think of roots as the points where the graph of the function crosses or touches the x-axis.
These points are crucial because they help us understand more about the behavior and characteristics of the graph.

• If you have the function \( f(x) = x^4 - 1 \), and you find \( f(1) = 0 \), this means that \( x = 1 \) is a root of the function.
• This point, \( x = 1 \), is where the graph of the function crosses the x-axis.
• Finding roots is especially important when solving equations, as it tells us possible solutions or values of \( x \) that satisfy the equation \( f(x) = 0 \).

Understanding the concept of roots helps you solve many types of problems in mathematics, especially those involving algebraic equations.

Evaluating Functions

Evaluating a function means finding the value of the function for a specific input value. This is a fundamental skill needed to understand how functions behave for different inputs.

• For the given function \( f(x) = x^4 - 1 \), if you need to find \( f(1) \), you evaluate by substituting \( x \) with 1 in the function.
• Evaluating gives you \( f(1) = 1^4 - 1 \), which is calculated as \( 1 - 1 = 0 \).
• This tells us the value of the function when \( x = 1 \) is 0, but it also provides insight into whether 1 is a root.

By frequently evaluating functions, you'll gain the ability to predict how changes in input affect the function's output, which is a powerful analytical tool in math and sciences.

Substitution Method

Substitution is a method often used to simplify the process of evaluating a function or solving an equation. Through substitution, you replace one variable with a given value to see how it affects the equation or expression.

• Apply substitution by identifying the variable, which is \( x \) in our function \( f(x) = x^4 - 1 \), and replacing it with the given value, \( a = 1 \).
• This involves changing the function from \( f(x) \) to \( f(1) \) and leads to substituting \( 1 \) into the equation, resulting in \( 1^4 - 1 \).
• By simplifying, you get a result that helps determine the characteristics of the equation, such as confirming whether a given point like \( x = 1 \) is a solution or root of the function.

The substitution method is not only reserved for solving for roots but is widely used in solving systems of equations and dealing with algebraic expressions, making it a versatile tool in mathematics.