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A function \( \mathrm{g}(\mathrm{x})=\mathrm{x}+2 \) and \( \mathrm{h}(\mathrm{x})=x^{2} \). Find the value of x for which \( \mathrm{hg}(\mathrm{x})=3 x^{2}+x-1 \) Select one: a. \( \quad x=\frac{-5}{2}, 1 \) b. \( \quad x=\frac{5}{2}, 1 \) c. \( \quad x=\frac{-5}{2},-1 \) d. \( \quad x=\frac{5}{2},-1 \)

Ask by Turnbull Rose. in Ghana
Mar 15,2025

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Answer

The solutions are \( x = \frac{5}{2} \) and \( x = -1 \), so the correct answer is option d.

Solution

We are given the functions \[ \mathrm{g}(\mathrm{x}) = \mathrm{x} + 2 \quad \text{and} \quad \mathrm{h}(\mathrm{x}) = \mathrm{x}^2. \] The composition is \[ \mathrm{hg}(\mathrm{x}) = \mathrm{h}(\mathrm{g}(\mathrm{x})) = \big(\mathrm{g}(\mathrm{x})\big)^2 = (\mathrm{x} + 2)^2. \] We need to find \( x \) such that \[ (\mathrm{x} + 2)^2 = 3\mathrm{x}^2 + \mathrm{x} - 1. \] **Step 1. Expand the left-hand side:** \[ (\mathrm{x} + 2)^2 = \mathrm{x}^2 + 4\mathrm{x} + 4. \] **Step 2. Set up the equation:** \[ \mathrm{x}^2 + 4\mathrm{x} + 4 = 3\mathrm{x}^2 + \mathrm{x} - 1. \] **Step 3. Rearrange the equation to form a quadratic equation:** Subtract \(\mathrm{x}^2 + 4\mathrm{x} + 4\) from both sides: \[ 0 = 3\mathrm{x}^2 + \mathrm{x} - 1 - (\mathrm{x}^2 + 4\mathrm{x} + 4). \] Simplify: \[ 0 = (3\mathrm{x}^2 - \mathrm{x}^2) + (\mathrm{x} - 4\mathrm{x}) + (-1 - 4), \] \[ 0 = 2\mathrm{x}^2 - 3\mathrm{x} - 5. \] **Step 4. Solve the quadratic equation \( 2\mathrm{x}^2 - 3\mathrm{x} - 5 = 0 \):** The quadratic formula is given by \[ \mathrm{x} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 2 \), \( b = -3 \), and \( c = -5 \). First, calculate the discriminant: \[ \Delta = b^2 - 4ac = (-3)^2 - 4(2)(-5) = 9 + 40 = 49. \] Now, compute \( x \): \[ \mathrm{x} = \frac{-(-3) \pm \sqrt{49}}{2 \cdot 2} = \frac{3 \pm 7}{4}. \] This gives two solutions: \[ \mathrm{x} = \frac{3 + 7}{4} = \frac{10}{4} = \frac{5}{2}, \] \[ \mathrm{x} = \frac{3 - 7}{4} = \frac{-4}{4} = -1. \] **Step 5. Identify the correct answer:** The two solutions are \( \mathrm{x} = \frac{5}{2} \) and \( \mathrm{x} = -1 \). Thus, the correct option is: \[ \text{d. } \quad x = \frac{5}{2}, -1. \]

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To find the value of \( x \) for which \( hg(x) = 3x^2 + x - 1 \), we first need to determine the composition \( hg(x) \). Here, \( g(x) = x + 2 \), so we can substitute this into \( h(x) \): \[ h(g(x)) = h(x + 2) = (x + 2)^2 \] Now, calculating \( (x + 2)^2 \): \[ (x + 2)^2 = x^2 + 4x + 4 \] Next, we set this equal to \( 3x^2 + x - 1 \): \[ x^2 + 4x + 4 = 3x^2 + x - 1 \] Now, rearranging the equation to set it to zero: \[ 0 = 3x^2 + x - 1 - (x^2 + 4x + 4) \] \[ 0 = 3x^2 + x - 1 - x^2 - 4x - 4 \] \[ 0 = 2x^2 - 3x - 5 \] Now, we can use the quadratic formula to solve for \( x \): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} \] Calculating the discriminant: \[ (-3)^2 - 4 \cdot 2 \cdot (-5) = 9 + 40 = 49 \] Now substituting back into the quadratic formula: \[ x = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4} \] This gives two solutions: \[ x = \frac{10}{4} = \frac{5}{2} \quad \text{and} \quad x = \frac{-4}{4} = -1 \] Thus, the values of \( x \) that satisfy the equation are \( x = \frac{5}{2}, -1 \). The correct answer is: d. \( \quad x=\frac{5}{2},-1 \)

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