Problem 23 - Algebra 2 and Trigonometry

Question

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=x^{-3} \cdot x^{4} ; \mathrm{f}(1) $$

Step-by-Step Solution

Answer

\( f(1) = 1 \)

Step 1: Simplify the Function

The function is given as \( f(x) = x^{-3} \cdot x^4 \). We can simplify this by applying the rule for multiplying powers with the same base: \( x^m \cdot x^n = x^{m+n} \). Therefore, \( f(x) = x^{-3 + 4} = x^1 \). Thus, \( f(x) = x \).

Step 2: Evaluate the Function at the Given Value

We need to evaluate the function \( f(x) \) at \( x = 1 \). Since \( f(x) = x \), substituting \( x = 1 \) into the function gives us: \( f(1) = 1 \).

Key Concepts

Exponent Rules

When working with exponents, several rules help us simplify and evaluate expressions. One key rule is when you multiply expressions with the same base, you add the exponents: \( x^m \cdot x^n = x^{m+n} \). This is called the product of powers property. For example, in the provided exercise, we had the expression \( x^{-3} \cdot x^{4} \). By applying the exponent rule, we add the exponents together:

• \( -3 + 4 = 1 \)

Hence, \( x^{-3} \cdot x^{4} = x^1 \), simplifying down the expression. Knowing these rules allows for efficient calculation and simplification, essential for tackling algebraic expressions accurately.

Simplifying Expressions

Simplifying expressions is about rewriting them in a form that is easier to work with or evaluate. It often involves combining like terms, expanding expressions, and applying rules such as the exponent rules.In our exercise, after applying the exponent rule, we simplified \( f(x) = x^{-3} \cdot x^{4} \) to \( f(x) = x^1 \).

• This reduced expression means the function can be directly evaluated, simplifying it significantly.

The goal is always to reduce the complexity of the expression without changing its value, which makes the evaluation much simpler.

Algebraic Functions

Algebraic functions are expressions that involve variables and constants combined using operations such as addition, subtraction, multiplication, division, and exponentiation. They are fundamental in mathematics for describing various relationships. In our exercise, the function \( f(x) = x^{-3} \cdot x^{4} \), once simplified, becomes a linear function \( f(x) = x \). Linear functions are among the simplest algebraic functions, represented as \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

• Evaluating algebraic functions involves substituting a given value for the variable, completing operations as specified by the function.
• For \( f(1) \), since \( f(x) \) simplifies to \( x \), the evaluation is straightforward: \( f(1) = 1 \).

Algebraic functions, when understood properly, lead to deeper insights and pave the way for more advanced mathematical concepts.