Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(g(t)=\pi-(t-2)^{2 / 3}\)

Step-by-Step Solution

Answer

The domain of the function \( g(t)=\pi-(t-2)^{2/3} \) is all real numbers, \( \mathbb{R} \).

Step 1: Understanding the Function

The function given is \( g(t) = \pi - (t-2)^{2/3} \). It is composed of two parts: the constant \( \pi \) and the variable expression \( (t-2)^{2/3} \). The \( 2/3 \) exponent indicates a power involving a cube root.

Step 2: Recognizing the Domain

To determine the domain of the function \( g(t) \), consider where the cube root expression \( (t-2)^{2/3} \) is defined. Cube roots are defined for all real numbers, so \( t \) can be any real number. Thus, the domain of \( g(t) \) is all real numbers, \( \mathbb{R} \).

Step 3: Analyzing the Structure

The constant \( \pi \) shifts the function vertically, and since none of the transitions involve undefined operations like division by zero, the function is continuous for all \( t \). The domain check confirmed this behavior.

Step 4: Behavior Near Specific Points

Although the domain is all real numbers, investigate the behavior at specific points, like \( t=2 \), where the expression becomes \( \pi - 0 = \pi \). Notice that as \( t \) becomes very large or very small, the function maintains continuity without reaching infinity.

Key Concepts

Power functions

Power functions are expressions that involve a variable raised to a certain power. In the case of the function \( g(t) = \, \pi - (t-2)^{2/3} \), the part \((t-2)^{2/3}\) is what we call a power function. It means that instead of being a simple linear increase or decrease, the variable affects the result in a more complex way. The power function here is raised to a fractional power, meaning it involves roots as well.

• Fractional Powers: When a power is a fraction, the numerator indicates the power, and the denominator represents the root. Thus, \( 2/3 \) implies square of the cube root.
• Transformation Effects: The \( (t-2) \) indicates a horizontal translation. In this context, the alteration happens two units to the right along the t-axis before applying any cube root transformations.

Understanding power functions helps us grasp how rapidly or slowly the output of a function changes as the input varies. In fractional powers, this change often appears less intuitive, as it connects with root-based operations.

Cube roots

Cube roots are the inverse operation of cubing a number. In the expression \( (t-2)^{2/3} \), the cube root is an inherent part of the calculation.

• All Real Numbers: Cube roots are defined for all real numbers, which means there are no restrictions like there are for even roots, such as square roots, where we cannot take the root of a negative number without complex results.
• Consistency in Sign: Unlike square roots, cube roots of negative numbers result in negative outcomes, maintaining the negative aspect throughout.

In our function, \( (t-2)^{2/3} \), after calculating the cube root, we square the result (as indicated by the numerator '2' in the fractional power). This operation neutralizes any negatives because a squared number, whether initially positive or negative, always becomes positive.

Function domain

The domain of a function tells us the set of allowable inputs. For the function \( g(t) = \pi - (t-2)^{2/3} \), determining the domain involves understanding where the expressions within are valid. Given cube roots are valid for all real numbers, \( t \) can take any real value from negative infinity to positive infinity.

• No Restrictions: Unlike functions involving square roots or division, where restrictions like non-negative radicands or non-zero denominators exist, our function has none.
• Real Number Domain: This essentially means our function can smoothly and continuously map any real number to a result without jumping or skipping values.

Recognizing domain aids in understanding where the function is applicable and ensures we're working within the boundaries of valid computation.

Continuity in functions

Continuity in a function implies no sudden jumps or breaks when graphing. For the function \( g(t) = \pi - (t-2)^{2/3} \), continuity derives from the fact that both components, \( \pi \) and \((t-2)^{2/3}\), are continuous.

• Everywhere Continuous: Since cube roots are continuous, \((t-2)^{2/3}\) doesn't disrupt the function. Coupled with the constant \( \pi \), the whole function remains unaffected by discontinuities.
• Behavior Near Specific Points: At points like \( t=2 \), where \( (t-2) \) becomes zero, the overall function result is simply \( \pi \), showcasing no erratic jumps or undefined values.

Continuity is crucial in calculus as it guarantees stability in a function's value as we make small changes to \( t \). For students, it means confidently applying calculus operations knowing no sudden loss of validity.