Question Find the domain of \( \mathrm{f}(\mathrm{x}) \). \[ f(x)=\sqrt{-x^{2}+5 x-4} \] Answer Attempt 1 out of 2 Answer Type: Interval watch Vic

To find the **domain** of the function \( f(x) = \sqrt{ -x^2 + 5x - 4 } \), we need to determine all real numbers \( x \) for which the expression inside the square root is **non-negative** (since the square root of a negative number is not a real number). ### Step 1: Set the Inside of the Square Root to be Non-Negative \[ -x^2 + 5x - 4 \geq 0 \] ### Step 2: Rearrange the Inequality It's often easier to work with positive leading coefficients. Multiply both sides of the inequality by -1, remembering to **reverse the inequality sign** when multiplying or dividing by a negative number: \[ x^2 - 5x + 4 \leq 0 \] ### Step 3: Factor the Quadratic Expression \[ x^2 - 5x + 4 = (x - 1)(x - 4) \] So the inequality becomes: \[ (x - 1)(x - 4) \leq 0 \] ### Step 4: Find the Critical Points The critical points are the values of \( x \) that make each factor zero: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] These points divide the real number line into three intervals: 1. \( x < 1 \) 2. \( 1 \leq x \leq 4 \) 3. \( x > 4 \) ### Step 5: Test Each Interval Choose a test point in each interval to determine where the inequality holds. 1. **Interval \( x < 1 \)**: - **Test Point**: \( x = 0 \) - \((0 - 1)(0 - 4) = (-1)(-4) = 4 > 0\) → Doesn't satisfy \( \leq 0 \) 2. **Interval \( 1 \leq x \leq 4 \)**: - **Test Point**: \( x = 2 \) - \((2 - 1)(2 - 4) = (1)(-2) = -2 < 0\) → Satisfies \( \leq 0 \) 3. **Interval \( x > 4 \)**: - **Test Point**: \( x = 5 \) - \((5 - 1)(5 - 4) = (4)(1) = 4 > 0\) → Doesn't satisfy \( \leq 0 \) ### Step 6: Include the Endpoints Check the critical points \( x = 1 \) and \( x = 4 \): - At \( x = 1 \): \[ f(1) = \sqrt{ -1^2 + 5(1) - 4 } = \sqrt{ -1 + 5 - 4 } = \sqrt{0} = 0 \quad \text{(Defined)} \] - At \( x = 4 \): \[ f(4) = \sqrt{ -4^2 + 5(4) - 4 } = \sqrt{ -16 + 20 - 4 } = \sqrt{0} = 0 \quad \text{(Defined)} \] ### Final Answer The function \( f(x) = \sqrt{ -x^2 + 5x - 4 } \) is defined for all real numbers \( x \) in the interval **[1, 4]**. \[ \boxed{\, [\,1,\ 4\,] \,} \]