Problem 63 - Algebra and Trigonometry

Question

61–68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = x^{2} + {x}$$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Define Even and Odd Functions

A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain. The function is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.

Step 2: Calculate f(-x)

Substitute \( -x \) for \( x \) in the function: \( f(-x) = (-x)^2 + (-x) = x^2 - x \).

Step 3: Compare f(-x) with f(x)

The original function is \( f(x) = x^2 + x \). We have \( f(-x) = x^2 - x \). Since \( f(-x) eq f(x) \) and \( f(-x) eq -f(x) \), the function is neither even nor odd.

Step 4: Sketch the Graph Conceptually

Although \( f(x) = x^2 + x \) is neither even nor odd, we can sketch its graph based on its components. The term \( x^2 \) represents a parabola opening upwards, and \( x \) shifts the parabola linearly, creating an asymmetrical curve.

Key Concepts

Function Symmetry

Function symmetry is a way to determine how a function behaves when you change the signs of its inputs. It's a valuable characteristic that can tell you a lot about the function's equation and graph, even before plotting it. Functions can be symmetric in two main ways: even symmetry or odd symmetry. An even function, such as cosine, mirrors itself across the y-axis. Mathematically, a function is even if for every input \( x \), the function holds where \( f(-x) = f(x) \). On the other hand, odd functions, such as sine, are symmetric with respect to both the origin in their graph—which means a half-turn will bring the function's graph to match with itself. For odd functions, the hallmark is \( f(-x) = -f(x) \). If a function doesn't meet any of these conditions, it is neither even nor odd, and its graph doesn't exhibit these types of symmetry. Understanding whether a function is even or odd can help simplify problems and make predictions about its graph.

Graph Sketching

Graph sketching allows us to visualize what the function looks like and understand its overall behavior. When sketching the graph of a function, several key factors need attention:

• Identify the symmetry of the function, if any, since this can simplify the sketching process by knowing the expected symmetry on the graph.
• Find where the function intersects the x-axis and y-axis, which provides a framework for the graph.
• Consider the end behavior of the function, which describes how the function behaves as \( x \) moves towards positive or negative infinity.

For functions like \( f(x) = x^2 + x \), understanding that it's neither even nor odd aids in knowing that its graph won't exhibit symmetry across the y-axis or around the origin. Since the squared term \( x^2 \) creates a parabolic effect while the linear term \( x \) linearly shifts this effect, the combination results in an asymmetrical graph that still retains a predictable shape—opening upwards due to the positive leading coefficient.

Polynomial Functions

Polynomial functions are an entire class of functions that consist of variables raised to whole number powers, with coefficients attached. They can vary from just a constant to high-degree polynomials with rich and intricate graphs. Key points about polynomials include:

• The degree of the polynomial determines the most terms in its expression and largely dictates its end behavior.
• Positive leading coefficients mean the function will rise to infinity as \( x \) becomes large, and a negative coefficient conversely means the function will fall to infinity.
• Roots or solutions of the polynomial (where the function equals zero) are highly informative, as these points correspond to x-intercepts of the graph.

In the specific case of \( f(x) = x^2 + x \), the function is a degree 2 polynomial, also known as a quadratic polynomial. Such functions typically form a parabolic shape. This polynomial lacks symmetry due to the presence of the linear \( x \) term, which moves the graph and alters the symmetry conditions usually seen in pure even or odd functions.