To understand the function [tex]\( f(x) = \sin^2(x) + 2 \sin(x) + 10 \)[/tex], let's break it down step-by-step.
### Step 1: Identify and Understand Each Part of the Function
1. [tex]\( \sin(x) \)[/tex]: This is the sine function, which is periodic with period [tex]\(2\pi\)[/tex] and varies between -1 and 1.
2. [tex]\( \sin^2(x) \)[/tex]: Squaring the sine function means that the range is transformed to [0, 1] since squaring any real number results in a non-negative result.
3. [tex]\( 2 \sin(x) \)[/tex]: This term scales the sine function by 2. Its range becomes [-2, 2].
4. 10: This is a constant term.
### Step 2: Combine the Terms
Given the function [tex]\( f(x) = \sin^2(x) + 2 \sin(x) + 10 \)[/tex]:
- Range of [tex]\( \sin^2(x) \)[/tex]: As mentioned, [tex]\( \sin^2(x) \in [0, 1] \)[/tex].
- Range of [tex]\( 2 \sin(x) \)[/tex]: As mentioned, [tex]\( 2 \sin(x) \in [-2, 2] \)[/tex].
When we add these together:
- The minimum value of [tex]\( \sin^2(x) + 2 \sin(x) \)[/tex] occurs when [tex]\( \sin(x) = -1 \)[/tex]. Substituting, we get [tex]\( (-1)^2 + 2(-1) = 1 - 2 = -1 \)[/tex].
- The maximum value of [tex]\( \sin^2(x) + 2 \sin(x) \)[/tex] occurs when [tex]\( \sin(x) = 1 \)[/tex]. Substituting, we get [tex]\( (1)^2 + 2(1) = 1 + 2 = 3 \)[/tex].
### Step 3: Add the Constant Term
Given we have [tex]\( \sin^2(x) + 2 \sin(x) \)[/tex] ranging from -1 to 3, adding the constant term 10 shifts this range up by 10:
- The minimum value of [tex]\( \sin^2(x) + 2 \sin(x) + 10 \)[/tex] is [tex]\(-1 + 10 = 9\)[/tex].
- The maximum value of [tex]\( \sin^2(x) + 2 \sin(x) + 10 \)[/tex] is [tex]\(3 + 10 = 13\)[/tex].
### Step 4: Conclusion
Therefore, the function [tex]\( f(x) = \sin^2(x) + 2 \sin(x) + 10 \)[/tex] ranges from 9 to 13. By evaluating the function [tex]\( f(x) \)[/tex], we see that it is already simplified. The final form of the function is:
[tex]\[ f(x) = \sin^2(x) + 2 \sin(x) + 10 \][/tex]
