Question

True or False The function \(f(x)=x^{4}+x^{2}+x\) is an even function. Justify your answer.

Short answer

False, the function \(f(x)=x^{4}+x^{2}+x\) is not an even function.

Step-by-step solution

Substituting -x into the Function

Find the function, \(f(-x)\) by substituting \(-x\) in place of \(x\) in the given function. Therefore, \(f(-x) = (-x)^{4}+(-x)^{2}+(-x)\) = \(x^4 + x^2 - x\)

Comparing f(-x) and f(x)

Compare the function obtained in Step 1, \(f(-x) = x^4 + x^2 - x\), with the original function \(f(x)=x^{4}+x^{2}+x\). This helps us determine if \(f(x)\) equals \(f(-x)\).

Conclusion

Based on the comparison in step 2, it can be understood that \(f(x) ≠ f(-x)\) as the term \(x\) in \(f(x)\) becomes \(-x\) in \(f(-x)\). Therefore, the given function \(f(x) = x^{4}+x^{2}+x\) is not an even function.

Key concepts

Symmetry of Functions

Understanding symmetry of functions is like looking into a mathematical mirror. Just like how symmetry in shapes divides them neatly into equal halves, in functions, it tells us if certain parts of the function's graph mimic each other when folded over a line. There are two primary types of symmetry to be aware of: even symmetry and odd symmetry. A graph has even symmetry if it looks the same when reflected over the y-axis — imagine folding it along the y-axis and the two sides align perfectly. On the other hand, an odd symmetry exists when a function's graph is invariant under a half-turn rotation about the origin, meaning that if the graph is rotated 180 degrees, it appears unchanged. As for our exercise, when we substitute \-x\ into \(f(x)\) and the resulting expression doesn't match the original \(f(x)\), it indicates that the graph does not have even symmetry, leading us to conclude that the function can't be even.

Recognizing these symmetries can be essential. It simplifies understanding the nature of the function and informs us about its behavior, such as predicting the possible roots or zeroes of the function without plotting the whole graph.

Algebraic Manipulation

The term algebraic manipulation refers to the process of reshaping algebraic expressions into different forms, usually to make them easier to work with or to reveal some of their properties. Algebraic manipulation includes a broad collection of methods like factoring, expanding, simplifying, and substituting values. This process is pivotal in identifying the characteristics of functions like parity. In our original exercise, we engaged in algebraic manipulation by substituting \-x\ into the function \(f(x)\), resulting in a new expression that could then be compared to the initial function. Through this comparison, we can determine if a function is even, odd, or neither. The ability to artfully juggle and transform algebraic expressions is thus not just a handy test for function parity but a foundational skill in problem-solving across various areas of mathematics.

Function Parity

The concept of function parity boils down to whether a function's signs stay the same or change when inputting \(-x\) instead of \(x\). Even functions are symmetric with respect to the y-axis, which implies that \(f(x) = f(-x)\) for all \(x\) in the function's domain. Examples include \(x^2\) and \(cos(x)\). Meanwhile, odd functions have rotational symmetry around the origin and satisfy the condition \(f(-x) = -f(x)\) for all \(x\). Classic examples here are \(x^3\) and \(sin(x)\).

Real-life doesn't always give us purely even or odd functions, though. Plenty of functions are neither, which is precisely what we encountered in the given exercise where the function \(f(x)=x^{4}+x^{2}+x\) does not exhibit the properties required to be considered even. Grasping function parity helps in calculus, as it affects the integration of the function over symmetric intervals and can inform the behavior of series expansions.

Even Function Properties

Dive into the characteristics of even function properties, and you'll discover that they have a distinct set of features that set them apart. A function is classified as even if, for every \(x\) in the domain, \(f(x) = f(-x)\). These functions graphically display symmetry about the y-axis. This reflective property is not just a visual curiosity—it has practical implications in analysis and solving equations. For example, an even function defined on an interval that is symmetric about the origin, like \([-a, a]\), can simplify certain integral calculations. However, the function in our exercise, \(f(x)=x^{4}+x^{2}+x\), fails this test because \(f(x)\) does not equal \(f(-x)\) due to the linear term \(x\), which changes sign when \(x\) is replaced with \(-x\). This is a clear indicator that the function does not hold the even function properties, and thus, it is not even.