Problem 57 - Thomas Calculus

Question

In Exercises \(47-58,\) say whether the function is even, odd, or neither. Give reasons for your answer. $$ h(t)=2 t+1 $$

Step-by-Step Solution

Answer

The function \( h(t) = 2t + 1 \) is neither even nor odd.

Step 1: Define Even and Odd Functions

A function is even if for every input \( t \), the condition \( h(t) = h(-t) \) holds true. Similarly, a function is odd if \( h(-t) = -h(t) \). If neither condition holds true, the function is neither even nor odd.

Step 2: Verify Even Function Condition

To check if \( h(t) = 2t + 1 \) is even, substitute \( -t \) into the function: \[ h(-t) = 2(-t) + 1 = -2t + 1. \] Compare this with \( h(t) = 2t + 1 \). Clearly, \( h(t) eq h(-t) \). This means the function is not even.

Step 3: Verify Odd Function Condition

To check if \( h(t) = 2t + 1 \) is odd, we test if \( h(-t) \) equals \(-h(t)\). We have \[ -h(t) = -(2t + 1) = -2t - 1. \] Compare this with \( h(-t) = -2t + 1 \). Since \(-2t - 1 eq -2t + 1\), the function is not odd.

Step 4: Conclusion

Since the function \( h(t) = 2t + 1 \) is neither even nor odd (as it meets neither condition from Steps 2 and 3), we conclude that \( h(t) \) is neither even nor odd.

Key Concepts

Function Properties

Understanding function properties is crucial for analyzing the behavior of mathematical functions. There are a few key attributes of functions that we often need to verify, including whether they are even, odd, or neither.

• An even function has symmetry about the y-axis. Mathematically, this means for a function \( f(x) \), we require \( f(x) = f(-x) \) for all \( x \) within its domain.
• An odd function has rotational symmetry about the origin. This is expressed as \( f(-x) = -f(x) \) for every \( x \) value.
• A function might be neither if it does not meet the criteria for being even or odd.

To determine the nature of a function, substitute \(-x\) into the function and see if either condition holds. If neither condition is satisfied, the function is neither even nor odd. Grasping these properties helps in sketching graphs and solving more complex mathematical problems.

Mathematical Reasoning

Mathematical reasoning enables us to evaluate and verify the characteristics of functions rigorously. In the case of determining if a function is even, odd, or neither, logical analysis is essential. Firstly, we establish a premise—in this case, substituting \(-x\) into the function and comparing outputs. Through deductive reasoning, we move step by step:- **Step 1:** Check the even condition. Evaluate \( h(-x) \) and see if it equals \( h(x) \). If true, the function is even.- **Step 2:** Check the odd condition. Establish if \( h(-x) = -h(x) \). If true, confirm the function as odd.In our example of \( h(t) = 2t + 1 \), neither condition is satisfied, because: - \( h(-t) = -2t + 1 eq 2t + 1 \). - \( -h(t) = -2t - 1 eq -2t + 1 \).Therefore, by confirming the logical steps, we conclude the function is neither even nor odd. This systematic approach ensures decisions are based on consistent logic.

Polynomial Functions

Polynomial functions are expressions formed by coefficients and variables raised to whole number powers. They often appear in abstract algebra and calculus due to their straightforward computational nature. Examples include quadratic, cubic, and higher degree polynomials. The function \( h(t) = 2t + 1 \) is a simple linear polynomial, characterized by terms up to the first degree. Linear polynomials have unique properties:- They graph as straight lines with a constant slope. - They typically do not exhibit even or odd properties unless centered around the y-axis or origin.When analyzing the symmetry of polynomial functions, observe the powers of their terms:- Even polynomial terms (like \( x^2 \), \( x^4 \)) contribute to even functions.- Odd polynomial terms (like \( x \), \( x^3 \)) lead to odd functions.- Mixing even and odd powers usually results in a function that is neither.In \( h(t) = 2t + 1 \), the mixture of a first degree term \( 2t \) and constant \( 1 \) disrupts any potential symmetry, explaining why it is neither even nor odd. Recognizing these forms supports understanding and predicting function behavior visually and analytically.