Sure, let's evaluate the function [tex]\( f(x) = 3 - \sqrt{\frac{1}{2}} x + 11 \)[/tex] at [tex]\( x = 10 \)[/tex] step by step.
1. Identify the terms in the function:
The function is composed of three parts:
- A constant term: [tex]\( 3 \)[/tex]
- A term involving [tex]\( x \)[/tex]: [tex]\( -\sqrt{\frac{1}{2}} x \)[/tex]
- Another constant term: [tex]\( 11 \)[/tex]
2. Evaluate the constant terms:
These terms are independent of [tex]\( x \)[/tex]:
- The first constant term is [tex]\( 3 \)[/tex]
- The second constant term is [tex]\( 11 \)[/tex]
3. Evaluate the term involving [tex]\( x \)[/tex]:
We need to evaluate [tex]\( -\sqrt{\frac{1}{2}} x \)[/tex] at [tex]\( x = 10 \)[/tex]:
- Compute the coefficient: [tex]\(\sqrt{\frac{1}{2}}\)[/tex]. This simplifies to approximately [tex]\( 0.7071067811865475 \)[/tex].
- Multiply this by 10: [tex]\(\sqrt{\frac{1}{2}} \times 10 \approx 7.0710678118654755 \)[/tex].
- The term becomes [tex]\( -7.0710678118654755 \)[/tex] (since it is subtracted).
4. Combine all terms:
We now combine the evaluated constant terms and the term involving [tex]\( x \)[/tex]:
- The first constant term: [tex]\( 3 \)[/tex]
- The evaluated term involving [tex]\( x \)[/tex]: [tex]\( -7.0710678118654755 \)[/tex]
- The second constant term: [tex]\( 11 \)[/tex]
5. Sum these values:
[tex]\[
f(10) = 3 + 11 - 7.0710678118654755
\][/tex]
- Adding [tex]\( 3 \)[/tex] and [tex]\( 11 \)[/tex] gives us [tex]\( 14 \)[/tex].
- Subtracting [tex]\( 7.0710678118654755 \)[/tex] from [tex]\( 14 \)[/tex]:
[tex]\[
f(10) = 14 - 7.0710678118654755 \approx 6.9289321881345245
\][/tex]
Thus, the evaluated value for [tex]\( f(10) \)[/tex] is [tex]\( 6.9289321881345245 \)[/tex].
