Math  /  Algebra

QuestionRecall domain can be defined as "the set of input values for a function, which produce valid output values".
Give the domain of p(x)=x6+x34x2+3x+5p(x)=x^{6}+x^{3}-4 x^{2}+3 x+5 in interval notation. \square

Studdy Solution

STEP 1

What is this asking? What numbers can we plug into this equation without breaking any math rules? Watch out! Sometimes tricky functions have hidden gotchas like dividing by zero or taking the square root of a negative number, but not this time!

STEP 2

1. Analyze the function.
2. Determine the domain.

STEP 3

Alright, let's **analyze** this function!
We've got p(x)=x6+x34x2+3x+5p(x) = x^6 + x^3 - 4x^2 + 3x + 5.
This is a **polynomial**, which is just a fancy way of saying it's a sum of terms where xx is raised to some whole number power.

STEP 4

Think about what kinds of numbers we can plug in for xx.
Can we use **positive** numbers?
Sure! How about **negative** numbers?
Absolutely! What about **zero**?
Yep, that works too!

STEP 5

There are no fractions with xx in the denominator, so we don't have to worry about dividing by **zero**.
There are no square roots (or any even roots) of xx, so we don't have to worry about taking the square root of a **negative** number.

STEP 6

Since we can plug in any real number for xx without breaking any math rules, the **domain** of p(x)p(x) is all real numbers.

STEP 7

How do we write "all real numbers" in **interval notation**?
We write it as (,)(-\infty, \infty).
This means the domain includes every number from negative infinity to positive infinity!

STEP 8

The domain of p(x)=x6+x34x2+3x+5p(x) = x^6 + x^3 - 4x^2 + 3x + 5 is (,)(-\infty, \infty).

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