Problem 81 - A Graphical Approach to College Algebra

Question

Let \(f(x)=x^{3} .\) Evaluate each expression. $$f(-2)$$

Step-by-Step Solution

Answer

\(-8\)

Step 1: Identify the Function

The function given is \(f(x) = x^3\). This means that wherever \(x\) appears, we will raise it to the power of 3.

Step 2: Substitute the Value

Substitute \(x = -2\) into the function \(f(x) = x^3\). This means we calculate \((-2)^3\).

Step 3: Calculate the Power

To find \((-2)^3\), multiply \(-2\) by itself two more times: \[(-2) \times (-2) \times (-2) = -8\]

Step 4: Conclusion

The result of \(f(-2) = (-2)^3\) is \(-8\).

Key Concepts

Function Evaluation

Function evaluation is like playing a game of substitution. When you're given a function, the goal is to find what the function tells us for a specific input. In the exercise above, the function is defined as \(f(x) = x^3\), which is a rule that tells us how to convert an input \(x\) into an output. It's a mathematical relationship where for each input \(x\), we find the output by calculating \(x^3\).

To evaluate the function for a specific value of \(x\), such as \(-2\), we substitute the value into the formula where \(x\) is. It's like plugging in the numbers in a formula to see what result you get. Once you replace \(x\) with \(-2\), you're left with \((-2)^3\). This process helps us find the output of the function given any particular input.

Exponentiation

Exponentiation is a fundamental concept in algebra, involving raising numbers to a power. It might seem tricky, but it really means multiplying a number by itself a certain number of times. In our scenario, we deal with \((-2)^3\). Here, the number \(-2\) is the base, and \(3\) is the exponent. This notation tells us to multiply \(-2\) by itself three times.

There are some simple steps to follow for exponentiation:

• The base is the number we want to multiply. In this case, it's \(-2\).
• The exponent tells us how many times to use the base as a factor. Here, that number is 3.
• Calculate by multiplying: \((-2) \times (-2) \times (-2) = -8\).

Every time you multiply with an odd exponent, you'll keep the original sign of the base if it's negative, resulting in a negative product.

Cubic Functions

A cubic function is a type of polynomial where the highest degree is three, like \(f(x) = x^3\). This indicates that the function involves the cube or the third power of \(x\). Cubic functions have unique characteristics, such as their graphs, which can have curves and inflection points, making them more dynamic compared to linear or quadratic functions.

To understand cubic functions better, consider these properties:

• Unlike linear functions, which are straight lines, cubic functions curve, offering a more complex path.
• They can cross the x-axis up to three times, giving them a wavy shape.
• Cubic functions always have a single cube term that dominates the graph, such as \(x^3\) in our example.

In our exercise, because of the simple form \(f(x) = x^3\), the evaluation for any \(x\) is straightforward, but cubic functions in general can model more complex real-world phenomena such as volume calculations or certain types of growth.