Certainly! Let's go through the detailed steps to arrive at the given expression for [tex]\( y \)[/tex]:
1. Identify the variable and the function:
- We are given a function [tex]\( y \)[/tex] that depends on the variable [tex]\( x \)[/tex].
2. Construct the expression:
- We are constructing polynomial expressions. Specifically, polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
3. Form the function [tex]\( y \)[/tex]:
- In this case, the function [tex]\( y \)[/tex] is defined as:
[tex]\[
y = x^3 + 2
\][/tex]
Here’s a breakdown of the expression:
- [tex]\( x^3 \)[/tex]: This term represents [tex]\( x \)[/tex] raised to the power of 3. It is a cubic term and dictates that the function is a cubic polynomial.
- [tex]\( + 2 \)[/tex]: This is a constant term added to the cubic term. Constants are values that do not change.
So, we have:
[tex]\[ y = x^3 + 2 \][/tex]
This expression [tex]\( y = x^3 + 2 \)[/tex] is now our function, which defines [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
If [tex]\( x \)[/tex] is known, you can compute [tex]\( y \)[/tex] by cubing the value of [tex]\( x \)[/tex] and then adding 2 to the result. For instance:
- If [tex]\( x = 1 \)[/tex], then [tex]\( y = 1^3 + 2 = 1 + 2 = 3 \)[/tex].
- If [tex]\( x = 2 \)[/tex], then [tex]\( y = 2^3 + 2 = 8 + 2 = 10 \)[/tex].
This concludes our detailed examination of the expression [tex]\( y = x^3 + 2 \)[/tex].
