Question

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ g(t)=\left\\{\begin{array}{ll} 1 & \text { if } t \leq 0 \\ t+1 & \text { if } 0

Short answer

The function is neither even nor odd.

Step-by-step solution

Understand Function Conditions

The function \( g(t) \) is piecewise, defined by different expressions in different intervals of \( t \).1. \( g(t) = 1 \) for \( t \leq 0 \).2. \( g(t) = t + 1 \) for \( 0 < t < 2 \).3. \( g(t) = t^2 - 1 \) for \( t \geq 2 \).

Check for Evenness

A function is even if \( g(t) = g(-t) \) for all \( t \).- For \( t \leq 0 \): \( g(t) = 1 \), so \( g(-t) \) needs checking in the positive region where \( -t \) is...- For \( 0 < t < 2 \): no symmetric interval in negative; hence, cannot match \( t+1 \).- For \( t \geq 2 \): \( g(t) = t^2 - 1 \) and \( g(-t) = (-t)^2 - 1 = t^2 - 1 \), but only holds for this section alone.Thus, the function is not even as symmetry about the y-axis does not hold entirely.

Check for Oddness

A function is odd if \( g(-t) = -g(t) \) for all \( t \).- For \( t \leq 0 \), \( g(t) = 1 \) and must compare against \( -g(-t) 1\).- For \( 0 < t < 2 \), \( g(t) = t + 1 \) does not result in symmetric negatives: \( 2 > t > 0 \).- For \( t \geq 2 \), symmetry would need \( -1 - t^2 \), which doesn't occur.The function is not odd as it does not satisfy the condition for any interval completely.

Conclusion

Since the function neither satisfies the properties necessary to be even nor those necessary to be odd, the function is neither even nor odd.

Sketch the Graph

Draw the function:- At \( t \leq 0 \): \( g(t) = 1 \). This is a horizontal line along the x-axis at y = 1.- At \( 0 < t < 2 \): \( g(t) = t + 1 \). This is a straight line starting from \( (0,1) \) passing through \( (1,2) \) and ending just before \( (2,3) \).- At \( t \geq 2 \): \( g(t) = t^2 - 1 \). This is a parabola opening upwards starting from \( (2,3) \).These parts denote changes in behavior and various slopes or curves in the graph as \( t \) changes.

Key concepts

Even and Odd Functions

The concept of even and odd functions is essential in mathematics as it revolves around the symmetry of functions. A function is termed "even" if it satisfies the condition \( f(x) = f(-x) \) for all \( x \). In simpler terms, the graph of an even function is symmetric about the y-axis.

An "odd" function, conversely, fulfills \( f(-x) = -f(x) \) for all \( x \). This means its graph is symmetric with respect to the origin, often appearing as a reflection. Understanding these definitions is key because the symmetry aids in predicting and understanding the function's behavior without having to graph it entirely.

In our exercise, the function \( g(t) \) doesn't align with these symmetrical behaviors. Due to its piecewise nature, it lacks a consistent symmetry, neither satisfying the even nor odd conditions. This highlights how piecewise functions can behave distinctly across different intervals.

Graph Sketching

Graph sketching is a visual representation of a function, providing insight into its behavior over different domains. When tackling piecewise functions like our exercise, sketching becomes especially useful due to their segmented behaviors.

For the function \( g(t) \):

• The section where \( g(t) = 1 \) shows a constant function for \( t \leq 0 \), represented by a horizontal line along the x-axis at \( y = 1 \).
• From \( 0 < t < 2 \), the graph is a line \( g(t) = t + 1 \), starting from \( (0,1) \) and increasing linearly to just before \( (2,3) \).
• For \( t \geq 2 \), the parabola \( g(t) = t^2 - 1 \) takes shape, curving upwards starting at \( (2,3) \).

The sketch provides a clear visual aid, displaying changes in slope and curvature as \( t \) transitions across intervals. Using graph sketching, one can easily understand and communicate the characteristics of such multifaceted functions.

Function Symmetry

Function symmetry can simplify understanding the nature and behavior of functions. Symmetric functions are fundamental as they relate to even and odd functions, offering ways to deduce many properties at a glance.

When a function is symmetric about the y-axis, it is associated with even functions. This means for every point on one side of the y-axis, there is a mirrored point on the opposite side, simplifying calculation and prediction of the function's behavior.

Odd functions, symmetric about the origin, can be interpreted as when flipped 180 degrees, the graph appears unchanged. Such symmetry is a helpful tool for verifying and validating function properties without exhaustive calculation.

In our exercise, the lack of symmetry in \( g(t) \) due to its segmented nature shows the importance of checking each part separately. Each segment can behave differently, rendering the overall function neither symmetric nor easily categorized as even or odd.

Calculus Concepts

Calculus concepts like continuity and piecewise definitions are crucial in understanding functions like \( g(t) \). While the function isn't continuous across the entire real line, each segment within defined intervals is continuous and can be analyzed using calculus.

For instance, the straight line \( g(t) = t + 1 \) in the interval \( 0 < t < 2 \) is continuous and differentiable. It has a constant derivative that indicates a fixed slope. However, the transition points, such as \( t = 0 \) and \( t = 2 \), might not maintain continuity via common calculus approaches.

• At \( t = 0 \), the function jumps from 1 to \( t + 1 \), indicating a discontinuity.
• At \( t = 2 \), it transitions from \( 3 \) to starting a parabolic shape \( t^2 - 1 \).

Calculus aids in dissecting these behaviors and understanding how each independent segment transitions and behaves within its domain, essential for more complex function analysis.