Question
For the function \( f(x)=\sqrt{3-x} \), find \( f(0), f(-x),-f(x), f(x+1) \), and \( f(x+h) \) (Use symbolic notation and fractions where needed.)
Ask by Wheeler Reed. in the United States
Jan 29,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( f(0) = \sqrt{3} \), \( f(-x) = \sqrt{3 + x} \), \( -f(x) = -\sqrt{3 - x} \), \( f(x + 1) = \sqrt{2 - x} \), \( f(x + h) = \sqrt{3 - x - h} \).
Solution
Let's evaluate each expression step by step for the function \( f(x) = \sqrt{3 - x} \).
1. **\( f(0) \)**
\[
f(0) = \sqrt{3 - 0} = \sqrt{3}
\]
2. **\( f(-x) \)**
\[
f(-x) = \sqrt{3 - (-x)} = \sqrt{3 + x}
\]
3. **\( -f(x) \)**
\[
-f(x) = -\sqrt{3 - x}
\]
4. **\( f(x + 1) \)**
\[
f(x + 1) = \sqrt{3 - (x + 1)} = \sqrt{2 - x}
\]
5. **\( f(x + h) \)**
\[
f(x + h) = \sqrt{3 - (x + h)} = \sqrt{3 - x - h}
\]
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Simplify this solution Beyond the Answer
To find the different values of the function \( f(x) = \sqrt{3 - x} \), we can evaluate it for the given inputs. 1. \( f(0) \): \[ f(0) = \sqrt{3 - 0} = \sqrt{3} \] 2. \( f(-x) \): \[ f(-x) = \sqrt{3 - (-x)} = \sqrt{3 + x} \] 3. \( -f(x) \): \[ -f(x) = -\sqrt{3 - x} \] 4. \( f(x + 1) \): \[ f(x + 1) = \sqrt{3 - (x + 1)} = \sqrt{3 - x - 1} = \sqrt{2 - x} \] 5. \( f(x + h) \): \[ f(x + h) = \sqrt{3 - (x + h)} = \sqrt{3 - x - h} \] So the evaluations are: - \( f(0) = \sqrt{3} \) - \( f(-x) = \sqrt{3 + x} \) - \( -f(x) = -\sqrt{3 - x} \) - \( f(x + 1) = \sqrt{2 - x} \) - \( f(x + h) = \sqrt{3 - x - h} \)
