Problem 23 - The Calculus with Analytic Geometry

Question

\(f(x)=2+(x-3)^{1 / 3}\)

Step-by-Step Solution

Answer

The function is \( f(x) = 2 + (x-3)^{1/3} \). Its domain and range are all real numbers.

Step 1: Identify the Function

The given function is \( f(x) = 2 + (x-3)^{1/3} \).

Step 2: Determine the Domain of the Function

For \( (x-3)^{1/3} \) to be defined, any real number works as the cubic root function is defined for all real numbers. Therefore, the domain of \( f(x) \) is all real numbers, or \( (-fty, fty) \).

Step 3: Find the Range of the Function

The function inside the cube root, \( (x-3)^{1/3} \), can take any real value. Adding 2 does not restrict this, so the range of \( f(x) \) is also all real numbers, or \( (-fty, fty) \).

Step 4: Calculate f(3)

Plug \( x = 3 \) into the function: \( f(3) = 2 + (3-3)^{1/3} = 2 + 0 = 2 \).

Step 5: Discuss the Behavior of the Function

As \( x \) increases or decreases, \( (x-3)^{1/3} \) increases or decreases, respectively. The function increases without bound as \( x \) increases and decreases without bound as \( x \) decreases.

Key Concepts

domain of the function

Understanding the domain of a function helps us know all the possible input values the function can accept. For the function given, \( f(x) = 2 + (x-3)^{1/3} \), we focus on the cubic root part, \( (x-3)^{1/3} \). The cubic root function is defined for all real numbers. This means that no matter what real number we plug into \( (x-3) \), we will get a real number as a result. Therefore, the domain of our function is all real numbers. In mathematical notation, this is represented as \( (-\infty, \infty) \). This means you can choose any number from the set of real numbers, and the function will give you a corresponding output.

range of the function

The range of a function tells us the set of all possible output values. For the function \( f(x) = 2 + (x-3)^{1/3} \), let's first look at \( (x-3)^{1/3} \). Because the cubic root of any real number is defined and can be any real number, \( (x-3)^{1/3} \) will take on any value from \( -\infty \) to \( \infty \). When we add 2 to these values, it doesn't limit the output to a smaller set of values. Instead, it just shifts the entire set of values up by 2. So, the range of \( f(x) \) is also all real numbers. In mathematical terms, we express the range as \( (-\infty, \infty) \). This shows that the function can produce any real number as an output, no matter which real number you choose as \( x \).

behavior of the function

Analyzing the function behavior helps us understand how the function values change concerning the input values. For the function \( f(x) = 2 + (x-3)^{1/3} \), we observe the term \( (x-3)^{1/3} \). Here's what happens:
When \( x \) increases: As \( x \) becomes larger, say \( x = 10 \), the expression \( (x-3) \) becomes bigger, and since the cubic root function increases, \( (x-3)^{1/3} \) also grows. Adding 2 to this result means \( f(x) \) increases as \( x \) increases.
When \( x \) decreases: As \( x \) becomes smaller, say \( x = -5 \), the value of \( (x-3) \) becomes more negative. The cubic root of a negative number is negative, so \( (x-3)^{1/3} \) becomes more negative. Adding 2 means that \( f(x) \) will decrease.
Overall, as \( x \) increases toward positive infinity, \( f(x) \) increases without any upper limit. Conversely, as \( x \) decreases toward negative infinity, \( f(x) \) decreases without any lower limit.

solving equations involving the cubic root function

Solving equations is critical when working with functions. For the function \( f(x) = 2 + (x-3)^{1/3} \), we sometimes need to find particular values, like finding where \( f(x) \) equals a specific number.
Example: Solve \( f(x) = 5 \).
Step 1: Set the function equal to the given value: \( 2 + (x-3)^{1/3} = 5 \).
Step 2: Isolate the cubic root term: \( (x-3)^{1/3} = 3 \) (subtract 2 from both sides).
Step 3: Remove the cubic root by raising both sides to the power of 3: \( x-3 = 27 \).
Step 4: Solve for \( x \): \( x = 30 \).
This technique helps in finding specific input values for given outputs, allowing us to better understand the connections between inputs and outputs in the context of the cubic root function.