Problem 73 - College Algebra and Calculus: An Applied Approach

Question

State whether the function is even, odd, or neither. $$ f(x)=3 x^{4} $$

Step-by-Step Solution

Answer

The function \( f(x) = 3x^{4} \) is even.

Step 1: Substitute -x into the function

Begin by substituting \( -x \) for \( x \) in the function. Doing so means the function \( f(x) = 3x^{4} \) becomes \( f(-x) = 3(-x)^{4} \).

Step 2: Simplify the function

Simplify the expression by finding the result of \( (-x)^4 \). Remember that the exponent 4 here means (-x) is multiplied by itself four times. Since a negative multiplied by a negative gives a positive, the result is \( f(-x) = 3x^{4} \).

Step 3: Compare the simplified function with the original

Finally, compare the form of \( f(-x) \) with \( f(x) \). In this case, \( f(-x) = f(x) \), so the function is even.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to varying powers. In simple terms, these are functions made up of different terms, each of which contains a variable raised to a non-negative integer exponent and possibly multiplied by a coefficient. The general form of a polynomial function looks like this:

• \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\]

Here, the

• coefficients are \( a_n, a_{n-1}, \ldots, a_0 \)
• variable is \( x \)
• exponents are non-negative integers like \( n, n-1, \ldots \)

In the case of our specific function, \( f(x) = 3x^4 \), it's a polynomial with just one term, making it a monomial. The coefficient is 3 and the variable is raised to the power of 4. Polynomial functions are versatile and can model a wide range of real-world phenomena, from simple equations to intricate curves and graphs.

Symmetry in Functions

Symmetry in functions revolves around how graphs of functions behave under reflection. Two main types of symmetry are often discussed: even symmetry and odd symmetry, and they help determine the visual balance of functions on a coordinate plane.

**Even Symmetry:**

• A function \( f(x) \) is called even if its graph is symmetric about the y-axis.
• Mathematically, this means \( f(-x) = f(x) \) for all \( x \) in the domain.
• Common examples include quadratic functions like \( f(x) = x^2 \).

**Odd Symmetry:**

• A function is called odd if it is symmetric about the origin.
• This implies that \( f(-x) = -f(x) \).
• An example is the cubic function \( f(x) = x^3 \).

In our function \( f(x) = 3x^4 \), substituting \( -x \) does not change the function, indicating even symmetry. This means if you fold the graph over the y-axis, both halves match perfectly.

Determining Function Parity

Determining whether a function is even, odd, or neither is straightforward when you follow a methodical approach. To find the parity of a function, you typically perform these steps:

• **Step 1: Substitute -x into the function** - This helps to understand how the function changes if at all when the input is made negative.


• **Step 2: Simplify the function** - By simplifying the expression, you can observe if the terms change sign, remain the same, or do something else.


• **Step 3: Compare the simplified expression with the original function** - If \( f(-x) = f(x) \), the function is even. If \( f(-x) = -f(x) \), the function is odd. If neither condition holds, the function is neither.

For the function \( f(x) = 3x^4 \), substituting \( -x \) does not alter the expression, as described previously in the solution steps. This reveals the function to be even since \( f(-x) \) equals \( f(x) \). Understanding function parity is essential as it provides insights into graph shapes and symmetry properties.