To find [tex]\( f(x) \)[/tex] for the given function, let us analyze and understand its structure step-by-step:
1. Understanding the Function Components:
- The function [tex]\( f(x) \)[/tex] is composed of two parts:
- The first part is a square root, [tex]\( \sqrt{2x} \)[/tex].
- The second part is a polynomial term, [tex]\( 5x^2 \)[/tex].
2. Structure of the Function:
- The function can be written as:
[tex]\[
f(x) = \sqrt{2x} + 5x^2
\][/tex]
3. Breaking Down the Calculation:
- For any given [tex]\( x \)[/tex]:
- First, multiply [tex]\( x \)[/tex] by 2 to get [tex]\( 2x \)[/tex].
- Next, find the square root of [tex]\( 2x \)[/tex], which is [tex]\( \sqrt{2x} \)[/tex].
- Separately, calculate the square of [tex]\( x \)[/tex], which is [tex]\( x^2 \)[/tex], and then multiply by 5 to get [tex]\( 5x^2 \)[/tex].
4. Combining the Results:
- Add the result from the square root part and the polynomial part together. This sum will give us the value of the function [tex]\( f(x) \)[/tex].
5. Example Calculation:
- If you choose a specific value for [tex]\( x \)[/tex], you can follow the outlined steps to compute [tex]\( f(x) \)[/tex].
- For example, let [tex]\( x = 1 \)[/tex]:
1. Calculate [tex]\( 2 \cdot 1 = 2 \)[/tex].
2. Find [tex]\( \sqrt{2} \approx 1.414 \)[/tex].
3. Compute [tex]\( 1^2 = 1 \)[/tex] and then [tex]\( 5 \cdot 1 = 5 \)[/tex].
4. Add the results: [tex]\( 1.414 + 5 = 6.414 \)[/tex].
- Thus, [tex]\( f(1) \approx 6.414 \)[/tex].
6. General Formula:
- Summarizing, for any input value [tex]\( x \)[/tex], we use the general formula:
[tex]\[
f(x) = \sqrt{2x} + 5x^2
\][/tex]
By following these steps, you can evaluate the function [tex]\( f(x) \)[/tex] for any specific value of [tex]\( x \)[/tex].
