Math  /  Algebra

QuestionProvide an example that demonstrates that the following statement is not true. If the degree of a function is an even number, then the function is an even function.
The function f(x)=2x4+x32f(x)=2 x^{4}+x^{3}-2 has even degree but is not even. Substituting x-x for xx yields f(x)=2(x)4+(x)32=f(-x)=2(x)^{4}+(x)^{3}-2= \square . This is \square \square f(x)-f(x). so f(x)=2x4+x32f(x)=2 x^{4}+x^{3}-2 is not an even (Simplify your answers.)

Studdy Solution

STEP 1

What is this asking? Find an example of a function that has an even degree but isn't an even function.
Then, show *why* it isn't an even function. Watch out! Don't mix up even *functions* with even *degrees*!
They're related, but not the same thing!

STEP 2

1. Define the function
2. Test for evenness

STEP 3

We're given the function f(x)=2x4+x32f(x) = 2x^4 + x^3 - 2.
This is our **starting point**!
Notice the largest power of xx is **4**, which is an **even number**.
This means the function f(x)f(x) has an **even degree**.

STEP 4

To check if a function is even, we need to see if f(x)=f(x)f(-x) = f(x).
Let's **investigate**!

STEP 5

**Calculate** f(x)f(-x): f(x)=2(x)4+(x)32f(-x) = 2(-x)^4 + (-x)^3 - 2 f(x)=2(x4)x32f(-x) = 2(x^4) - x^3 - 2f(x)=2x4x32f(-x) = 2x^4 - x^3 - 2So, f(x)f(-x) simplifies to 2x4x322x^4 - x^3 - 2.

STEP 6

Now, let's **compare** f(x)f(-x) with f(x)f(x).
We have: f(x)=2x4+x32f(x) = 2x^4 + x^3 - 2 f(x)=2x4x32f(-x) = 2x^4 - x^3 - 2Notice the difference in the sign of the x3x^3 term!
Since the signs are different, f(x)f(-x) is *not* equal to f(x)f(x).

STEP 7

Let's also check if f(x)f(-x) is equal to f(x)-f(x).
We have: f(x)=(2x4+x32)-f(x) = -(2x^4 + x^3 - 2) f(x)=2x4x3+2-f(x) = -2x^4 - x^3 + 2Since f(x)=2x4x32f(-x) = 2x^4 - x^3 - 2 and f(x)=2x4x3+2-f(x) = -2x^4 - x^3 + 2, we can see that f(x)f(-x) is *not* equal to f(x)-f(x) either!

STEP 8

f(x)f(-x) simplifies to 2x4x322x^4 - x^3 - 2.
This is *not* equal to f(x)f(x), which is 2x4+x322x^4 + x^3 - 2.
This is also *not* equal to f(x)-f(x), which is 2x4x3+2-2x^4 - x^3 + 2.
Therefore, f(x)f(x) is *not* an even function, even though it has an even degree!

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