Problem 58 - 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 is even because \( h(-t) = h(t) \).

Step 1: Understand Even and Odd Functions

An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \) in the domain. An odd function satisfies \( f(-x) = -f(x) \). If neither condition holds, the function is neither even nor odd.

Step 2: Apply the Condition for Even Function

To check if the function \( h(t) = 2|t| + 1 \) is even, we calculate \( h(-t) \) and see if it equals \( h(t) \). Here, \( h(-t) = 2|-t| + 1 = 2|t| + 1 = h(t) \). Thus, \( h(t) \) is even.

Step 3: Confirm the Result

Since \( h(-t) = h(t) \), the function meets the criteria of an even function. Therefore, \( h(t) = 2|t| + 1 \) is an even function.

Key Concepts

Even Function

An even function is quite special because it maintains symmetry around the y-axis. When a function is even, it satisfies an important equation: \( f(-x) = f(x) \) for every value of \( x \) within its domain.
This means if you plug in a negative \( x \), the output value remains the same as plugging in the positive \( x \).
Think of pulling a blanket snugly over a straight line—the sides should mirror each other perfectly along the y-axis. In practical terms, this means an even function doesn't change its output value no matter whether you input a positive or negative \( x \).

Why is this rule significant?

• It helps identify the function's type quickly.
• It indicates symmetry properties, which are useful in graphing.
• It can aid in solving various mathematical problems efficiently.

In our example with the function \( h(t) = 2|t| + 1 \), when we replaced \( t \) with \( -t \), the expression remained unchanged, proving it is indeed an even function.

Function Symmetry

Symmetry in functions can make them easier to understand and work with, especially when graphing or analyzing their properties.
In the context of even and odd functions, symmetry plays a crucial role.

Functions exhibit two main types of symmetry:

• **Y-axis Symmetry:** This occurs when a function is even. The graph looks identical on both sides of the y-axis. If you fold the graph along the y-axis, both halves would match perfectly.
• **Origin Symmetry:** This occurs when a function is odd. If you rotate the graph 180 degrees around the origin, it will look the same.

When a function is neither even nor odd, it doesn’t exhibit these symmetrical properties, making them sometimes a bit trickier to graph. In our example, the function \( h(t) = 2|t| + 1 \) shows y-axis symmetry, making it an even function and easier to visualize accordingly.

Absolute Value Function

The absolute value function is a fascinating function because it concerns the distance from zero, hence always yields non-negative values. When you see \( |x| \), it simply means "how far is x from zero on the number line?," ignoring any negative sign.
This concept can be very useful, especially when dealing with real-world problems that can't have negative results, like distance.

In function terms, for any real number \( t \):

• \( |t| = t \) if \( t \geq 0 \)
• \( |t| = -t \) if \( t < 0 \)

In the function \( h(t) = 2|t| + 1 \), the absolute value ensures that no matter whether \( t \) is positive or negative, the outcome, \( 2|t| \), adjusts \( t \) to non-negative.
This property confirms why \( h(t) \) behaves as an even function, since \(|-t| = |t|\), yielding the same value regardless of the positive or negative sign of \( t \). Understanding how absolute value works can unravel many mathematics puzzles and provide assurance in handling both positive and negative inputs uniformly.