Problem 73 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the function \(f\) at the values \(f(-2), f(-1), f(0), f(1),\) and \(f(2) .\) \(f(x)=3^{x}\)

Step-by-Step Solution

Answer

The function evaluates to: \(f(-2) = \frac{1}{9}, f(-1) = \frac{1}{3}, f(0) = 1, f(1) = 3, f(2) = 9\).

Step 1: Evaluate f(-2)

The task is to evaluate the function at different values. Start with evaluating \(f(-2)\). Substitute \(-2\) into the function: \(f(-2) = 3^{-2}\). Simplify it to \(f(-2) = \frac{1}{3^2} = \frac{1}{9}\).

Step 2: Evaluate f(-1)

Next, substitute \(-1\) into the function: \(f(-1) = 3^{-1}\). Convert it: \(f(-1) = \frac{1}{3}\).

Step 3: Evaluate f(0)

Now substitute \(0\) into the function: \(f(0) = 3^0\). We know anything raised to the power of zero is one, so \(f(0) = 1\).

Step 4: Evaluate f(1)

Substitute \(1\) into the function: \(f(1) = 3^1\). Calculate to get \(f(1) = 3\).

Step 5: Evaluate f(2)

Finally, substitute \(2\) into the function: \(f(2) = 3^2\). Compute to get \(f(2) = 9\).

Key Concepts

Understanding Function Evaluation

Function evaluation is like following a recipe where you replace the ingredients with numbers. In mathematics, a function represents a relationship where each input is linked to exactly one output. To evaluate a function means to find the output when a specific input (or number) is plugged into the equation. For instance, consider the function given:

• If you have a function like \( f(x) = 3^x \), and you want to know what \( f(-2) \) is, you simply replace \( x \) with -2.
• So, \( f(-2) = 3^{-2} \).

Function evaluation is about finding an answer for the function at specific points using these substitutions. It's a step-by-step approach to uncover the results, which involves some critical thinking and careful substitution.

Grasping Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition or multiplication. These expressions can represent various quantities. For example, in the exercise, the expression is \( f(x) = 3^x \) where '3' is the base of the exponential function and 'x' is the variable. Working with algebraic expressions involves a few steps:

• Identify the components: the base (\(3\)) and the variable (\(x\)).
• Determine the operations: here, it involves an exponentiation operation \(3^x\).
• Simplify by substituting values for \(x\): replace \(x\) with different numbers to compute the function value.

Thorough understanding of algebraic expressions allows easier manipulation of mathematical formulas to solve problems effectively.

Understanding Power of a Number

The power of a number refers to the number of times you multiply it by itself. This is a fundamental concept in calculating exponential functions. In the equation \( f(x) = 3^x \), the number 3 is raised to various powers depending on the value of \(x\). Here’s how it works:

• The expression \( 3^1 \) is read as '3 to the power of 1', resulting in just 3.
• For \( 3^2 \), '3 to the power of 2' means 3 multiplied by itself, yielding 9.
• Conversely, \( 3^{-1} \) results in \( \frac{1}{3} \), because negative exponents represent the reciprocal.

This concept allows us to work with exponential expressions to determine their numerical values, which is essential in mathematics for analyzing relationships and growth patterns.

Mastering Exponent Rules

Exponent rules are guidelines that help simplify expressions involving powers of numbers. Knowing these rules is crucial for handling complex expressions. Here are some key rules used in the exercise explained with clarity:

• **Zero Exponent Rule:** Any non-zero number raised to the power of 0 equals 1, like \(3^0 = 1\).
• **Negative Exponent Rule:** When a number is raised to a negative exponent, it means you take the reciprocal of the number raised to the positive of that exponent, thus \(a^{-n} = \frac{1}{a^n}\).
• **Positive Exponents:** Raising a number to a positive power simply involves multiplying it by itself the indicated number of times, such as \(3^2 = 9\).

Grasping these rules makes it easier to evaluate expressions and solve problems involving exponents. They provide a straightforward method to diminish complexity in exponential expressions.