Sure! Let's go through the function [tex]\( f(x) \)[/tex] step-by-step to understand its structure and components. The function given is:
[tex]\[ f(x) = 3 x^2 \sin(4 x^2 + 1) + \log(4 x)^{2 x} \][/tex]
### Step-by-Step Solution:
1. Understanding the first part of the function:
[tex]\[
3 x^2 \sin(4 x^2 + 1)
\][/tex]
- Here, you have the term [tex]\( 3 x^2 \)[/tex], which is a simple quadratic term multiplied by 3.
- The term [tex]\( \sin(4 x^2 + 1) \)[/tex] is the sine of the expression [tex]\( 4 x^2 + 1 \)[/tex].
- Combining these two parts involves multiplying the quadratic term [tex]\( 3 x^2 \)[/tex] with the sine term [tex]\( \sin(4 x^2 + 1) \)[/tex].
2. Understanding the second part of the function:
[tex]\[
\log(4 x)^{2 x}
\][/tex]
- This part involves the logarithm function. The notation [tex]\( \log(4 x) \)[/tex] typically represents the natural logarithm (ln) of [tex]\( 4 x \)[/tex].
- The expression is then raised to the power [tex]\( 2 x \)[/tex].
- Therefore, you first compute the natural logarithm of [tex]\( 4 x \)[/tex] and then raise the result to the power [tex]\( 2 x \)[/tex].
3. Combining both parts:
Once you have both parts of the function understood, you combine them to get the final expression for [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 3 x^2 \sin(4 x^2 + 1) + \log(4 x)^{2 x}
\][/tex]
### Summary:
- The term [tex]\( 3 x^2 \sin(4 x^2 + 1) \)[/tex] involves a quadratic polynomial [tex]\( 3 x^2 \)[/tex] multiplied by the sine of a quadratic expression [tex]\( 4 x^2 + 1 \)[/tex].
- The term [tex]\( \log(4 x)^{2 x} \)[/tex] involves taking the natural logarithm of [tex]\( 4 x \)[/tex] and then raising the result to the power [tex]\( 2 x \)[/tex].
Hence, the function [tex]\( f(x) \)[/tex] in its simplified and combined form is:
[tex]\[
f(x) = 3 x^2 \sin(4 x^2 + 1) + \log(4 x)^{2 x}
\][/tex]
