Problem 17 - Brief Applied Calculus

Question

For each function: $$ f(x)=x^{2 / 3} ; \text { find } f(-8) $$

Step-by-Step Solution

Answer

The value of \( f(-8) \) is 4.

Step 1: Understand the given function

The function given is \( f(x) = x^{2/3} \). This function represents a number raised to the power of \( \frac{2}{3} \).

Step 2: Substitute \( x = -8 \)

To find \( f(-8) \), substitute \( -8 \) into the function for \( x \). So, we need to evaluate \( (-8)^{2/3} \).

Step 3: Simplify the expression \((-8)^{2/3}\)

The expression \((-8)^{2/3}\) can be evaluated by first finding the cube root of \(-8\), which is \(-2\), and then squaring the result. \((-8)^{2/3} = ( (-8)^{1/3} )^2\).

Step 4: Find \((-8)^{1/3}\)

The cube root of \(-8\) is \(-2\) because \((-2) \times (-2) \times (-2) = -8\).

Step 5: Square the result

Now, square \(-2\) to get \((-2)^2\).

Step 6: Calculate \((-2)^2\)

The expression \((-2)^2\) equals \(4\) because \((-2) \times (-2) = 4\).

Step 7: Write the final answer

Therefore, the value of \( f(-8) \) is \( 4 \).

Key Concepts

Fractional Exponents

Fractional exponents are a way to represent roots and powers together in one simple expression. Instead of writing the nth root of a number separately, fractional exponents combine it with a power. For example, the expression \[ x^{\frac{2}{3}} \] is an example of a fractional exponent. In this case, the base, "x", is taken to the power of 2/3. This means that you should find the cube root of the number first, which is written as \( x^{\frac{1}{3}} \), and then square that result. * The numerator "2" in the fraction \( \frac{2}{3} \) signifies the power. * The denominator "3" signifies the root type, in this case, the cube root. So, multiplying by a fractional exponent means involving both these operations in sequence. Understanding the sequence of operations is crucial because it impacts how you simplify expressions.

Cube Roots

Cube roots are used to reverse the process of cubing a number. When a number "x" is cubed, it means \( x \times x \times x \). The cube root is essentially the operation that gives you the original number when you have the result of cubing. In terms of notation, the cube root of a number \( x \) is written as \( x^{1/3} \). For our example, solving \((-8)^{1/3}\) means finding the cube root of \(-8\). We deduced that this is \(-2\) because \((-2) \times (-2) \times (-2) = -8\). When dealing with cube roots, remember:

• Cube roots can be negative because a negative number multiplied by itself three times is still negative.
• It's different from square roots, where the base number must be positive to result in a real number.

Understanding cube roots helps bridge the gap in evaluating more complex functions involving exponents.

Evaluating Functions

Evaluating a function is the process of finding the output value for a specific input value. To evaluate the function \( f(x) \), we substitute each "x" in the expression with a given number, then simplify to find the result.Let's consider the function \[ f(x) = x^{2/3} \] To find \( f(-8) \), we substituted \(-8\) for "x". That meant evaluating \[ (-8)^{2/3} \]. Following these steps:1. **Find the cube root of \(-8\).** We got \(-2\).2. **Square the result.** Squaring \(-2\) gave us \(4\).When evaluating functions:

• First, substitute the given value properly.
• Simplify step-by-step to avoid errors and gain accuracy.

This systematic method helps manage different types of expressions effectively, allowing accurate calculation of outcomes.