Problem 118 - Precalculus Essentials

Question

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. Prove that if \(f\) and \(g\) are even functions, then \(f g\) is also an even function.

Step-by-Step Solution

Answer

Yes, the statement is true. The proof shows that if \(f(x)\) and \(g(x)\) are even functions, the product \(f(x)g(x)\) also satisfies the condition of an even function, meaning that it is also even.

Step 1: Definition of Even Functions

Start by recalling the definition of an even function: A function \(f(x)\) is considered even if \(f(x) = f(-x)\) for all \(x\) in the function's domain. Similarly, a function \(g(x)\) is even if \(g(x) = g(-x)\). This will be the foundation of our proof.

Step 2: Express the Product of Two Functions

We are aiming to prove that the product \(h(x) = f(x)g(x)\) of two even functions is also even, i.e., \(h(x)=h(-x)\). Substituting the definition of an even function into the equation for \(h(-x)\), we get that \(h(-x) = f(-x)g(-x) = f(x)g(x) = h(x)\).

Step 3: Finalize the Proof

Since we have shown that the product function \(h(x) = f(x)g(x)\) satisfies the condition \(h(x) = h(-x)\) for all \(x\), by definition, \(h(x)\) is an even function. Therefore, we have proven that the product of two even functions is also even.

Key Concepts

Symmetry of Even Functions

Even functions have a very distinct characteristic: they are symmetrical about the y-axis. This means that for every point on the function with coordinates \( (x, y) \) there is a corresponding point with coordinates \( (-x, y) \) that also lies on the function. In simple terms, if you fold the graph along the y-axis, both halves would match exactly.

In a mathematical context, a function \( f \) is defined to be even if the value of \( f(x) \) is the same as \( f(-x) \) for every value of \( x \) in the function's domain. This can also be understood as a 'mirror effect' where the left side of the y-axis is a mirror image of the right side.

When it comes to the symmetry of even functions, visualization can help students grasp the concept. A common example of an even function is the quadratic function \( f(x) = x^2 \), where the parabolic graph clearly shows this symmetry. To reinforce the understanding, plotting several even functions and recognizing their symmetry can serve as a valuable exercise improvement advice.

Function Multiplication

When we talk about function multiplication, we are referring to taking two functions and producing a third one by multiplying them together. If you have two functions, \( f(x) \) and \( g(x) \) the product function is denoted as \( h(x) = f(x)g(x) \). The key thing to remember is that when you multiply two functions, the resulting function's value at \( x \) is simply the product of those two functions' values at the same \( x \).

This concept becomes especially interesting when combining certain types of functions. For instance, multiplying two even functions gives birth to another even function. This attribute can serve as a foundation for exploring more complex operations involving functions, such as division or composition. As an improvement, engaging in the practice of multiplying different pairs of functions - even, odd, or neither - could deepen students' understanding of function behavior under multiplication.

Proof in Mathematics

Proofs are the heart of mathematical certainty, providing a way to firmly establish the truth of a statement beyond any doubt. In a proof, we start with known truths, such as definitions, axioms, previously established theorems, and logical reasoning, to build a convincing argument that a certain mathematical statement is true.

For the proof at hand, which shows that the product of two even functions is even, we employed direct proof. This technique starts by assuming that the given conditions (in this case, that \( f \) and \( g \) are both even) are true, and then through logical deduction, it demonstrates that the conclusion follows (the product function \( h \) is also even).

To improve mathematical understanding, students can practice by identifying the type of proof used in different scenarios and by creating proofs of their own. Beginning with simpler statements and gradually tackling more complex ones, such as the proof that a certain function has an inverse, can foster a deeper appreciation for the structure and reasoning in mathematical arguments.