Alright, let's delve into the given function [tex]\( f(x) \)[/tex] and understand its various components step-by-step.
The function in question is:
[tex]\[ f(x) = \sqrt{2x} + 5x^2 \][/tex]
### Step-by-Step Solution:
1. Understand Each Term:
- The first term is [tex]\( \sqrt{2x} \)[/tex], which involves a square root of the product of 2 and [tex]\( x \)[/tex].
- The second term is [tex]\( 5x^2 \)[/tex], which is a basic polynomial term where [tex]\( x \)[/tex] is squared and then multiplied by 5.
2. Rewrite the Function:
Let's rewrite the function to clearly see its structure:
[tex]\[ f(x) = \sqrt{2x} + 5x^2 \][/tex]
3. Handling the Square Root Term:
Consider the first term [tex]\( \sqrt{2x} \)[/tex]:
- The expression [tex]\( 2x \)[/tex] is inside the square root. To manipulate or simplify this, remember that:
[tex]\[ \sqrt{2x} = \sqrt{2} \cdot \sqrt{x} \][/tex]
Thus, we separate it into two distinct square root factors.
4. Handling the Polynomial Term:
The second term [tex]\( 5x^2 \)[/tex] is already in its simplest form:
- [tex]\( 5 \)[/tex] is the coefficient.
- [tex]\( x^2 \)[/tex] is [tex]\( x \)[/tex] raised to the power of 2.
So, combining both observations, we can rewrite the function as:
[tex]\[ f(x) = \sqrt{2} \cdot \sqrt{x} + 5x^2 \][/tex]
However, this expression:
[tex]\[ \sqrt{2}\sqrt{x} + 5x^2 \][/tex]
is more commonly written in a slightly more compact form:
[tex]\[ f(x) = \sqrt{2x} + 5x^2 \][/tex]
### Conclusion
In summary, the function [tex]\( f(x) \)[/tex] combines a radical expression and a polynomial expression:
[tex]\[ f(x) = \sqrt{2} \cdot \sqrt{x} + 5x^2 \][/tex]
Given this function:
[tex]\[ f(x) = \sqrt{2x} + 5x^2 \][/tex]
we recognize it can also be written as:
[tex]\[ f(x) = \sqrt{2}\sqrt{x} + 5x^2 \][/tex]
stressing the separate nature of the constants and variables involved.
