For each of the following: i) Determine the equation of the inverse. ii) Draw the graph of the relation, its inverse, and the line \( y=x \). iii) Write down the domain and range for both the relation and its inverse \( \begin{array}{ll}\text { a) } f(x)=-\frac{3}{2} x^{2} & \text { b) } g(x)=-\frac{3}{2} x \\ \text { c) } h(x)=3 x & \text { d) } p(x)=3 x^{2}\end{array} \)

Rewrite the equation by following steps: - step0: Rewrite in standard form: \(y=3x\) - step1: Move the variable to the left side: \(y-3x=0\) - step2: Multiply both sides: \(-y+3x=0\) - step3: Reorder the terms: \(3x-y=0\) Rewrite the equation \( y=-\frac{3}{2}x \). Rewrite the equation by following steps: - step0: Rewrite in standard form: \(y=-\frac{3}{2}x\) - step1: Multiply both sides of the equation by LCD: \(2y=-3x\) - step2: Move the variable to the left side: \(2y+3x=0\) - step3: Reorder the terms: \(3x+2y=0\) Solve the equation \( y=3x \). Solve the equation by following steps: - step0: Solve for \(x\): \(y=3x\) - step1: Swap the sides: \(3x=y\) - step2: Divide both sides: \(\frac{3x}{3}=\frac{y}{3}\) - step3: Divide the numbers: \(x=\frac{y}{3}\) Solve the equation \( y=-\frac{3}{2}x \). Solve the equation by following steps: - step0: Solve for \(x\): \(y=-\frac{3}{2}x\) - step1: Swap the sides: \(-\frac{3}{2}x=y\) - step2: Change the signs: \(\frac{3}{2}x=-y\) - step3: Multiply by the reciprocal: \(\frac{3}{2}x\times \frac{2}{3}=-y\times \frac{2}{3}\) - step4: Multiply: \(x=-\frac{2}{3}y\) Solve the equation \( y=3x^{2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(y=3x^{2}\) - step1: Swap the sides: \(3x^{2}=y\) - step2: Divide both sides: \(\frac{3x^{2}}{3}=\frac{y}{3}\) - step3: Divide the numbers: \(x^{2}=\frac{y}{3}\) - step4: Simplify the expression: \(x=\pm \sqrt{\frac{y}{3}}\) - step5: Simplify: \(x=\pm \frac{\sqrt{3y}}{3}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{\sqrt{3y}}{3}\\&x=-\frac{\sqrt{3y}}{3}\end{align}\) Solve the equation \( y=-\frac{3}{2}x^{2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(y=-\frac{3}{2}x^{2}\) - step1: Swap the sides: \(-\frac{3}{2}x^{2}=y\) - step2: Change the signs: \(\frac{3}{2}x^{2}=-y\) - step3: Multiply by the reciprocal: \(\frac{3}{2}x^{2}\times \frac{2}{3}=-y\times \frac{2}{3}\) - step4: Multiply: \(x^{2}=-\frac{2}{3}y\) - step5: Simplify the expression: \(x=\pm \sqrt{-\frac{2}{3}y}\) - step6: Simplify: \(x=\pm \frac{\sqrt{-6y}}{3}\) - step7: Separate into possible cases: \(\begin{align}&x=\frac{\sqrt{-6y}}{3}\\&x=-\frac{\sqrt{-6y}}{3}\end{align}\) Let's analyze each function step by step to determine the inverse, graph the relations, and identify the domain and range. ### a) \( f(x) = -\frac{3}{2} x^{2} \) **i) Determine the equation of the inverse:** To find the inverse, we set \( y = -\frac{3}{2} x^{2} \) and solve for \( x \): \[ y = -\frac{3}{2} x^{2} \implies x = \pm \frac{\sqrt{-6y}}{3} \] Since \( f(x) \) is a downward-opening parabola, we restrict the domain to \( x \leq 0 \) to ensure the inverse is a function: \[ f^{-1}(y) = -\frac{\sqrt{-6y}}{3} \] **ii) Graph the relation, its inverse, and the line \( y=x \):** - The graph of \( f(x) \) is a downward parabola. - The inverse \( f^{-1}(y) \) is the left half of the upward parabola. - The line \( y=x \) is a diagonal line through the origin. **iii) Domain and Range:** - Domain of \( f(x) \): \( (-\infty, 0] \) - Range of \( f(x) \): \( (-\infty, 0] \) - Domain of \( f^{-1}(y) \): \( (-\infty, 0] \) - Range of \( f^{-1}(y) \): \( (-\infty, 0] \) --- ### b) \( g(x) = -\frac{3}{2} x \) **i) Determine the equation of the inverse:** To find the inverse, we set \( y = -\frac{3}{2} x \) and solve for \( x \): \[ y = -\frac{3}{2} x \implies x = -\frac{2}{3} y \] Thus, the inverse is: \[ g^{-1}(y) = -\frac{2}{3} y \] **ii) Graph the relation, its inverse, and the line \( y=x \):** - The graph of \( g(x) \) is a straight line with a negative slope. - The inverse \( g^{-1}(y) \) is also a straight line with a negative slope. - The line \( y=x \) is a diagonal line through the origin. **iii) Domain and Range:** - Domain of \( g(x) \): \( (-\infty, \infty) \) - Range of \( g(x) \): \( (-\infty, \infty) \) - Domain of \( g^{-1}(y) \): \( (-\infty, \infty) \) - Range of \( g^{-1}(y) \): \( (-\infty, \infty) \) --- ### c) \( h(x) = 3x \) **i) Determine the equation of the inverse:** To find the inverse, we set \( y = 3x \) and solve for \( x \): \[ y = 3x \implies x = \frac{y}{3} \] Thus, the inverse is: \[ h^{-1}(y) = \frac{y}{3} \] **ii) Graph the relation, its inverse, and the line \( y=x \):** - The graph of \( h(x) \) is a straight line with a positive slope. - The inverse \( h^{-1}(y) \) is also a straight line with a positive slope. - The line \( y=x \) is a diagonal line through the origin. **iii) Domain and Range:** - Domain of \( h(x) \): \( (-\infty, \infty) \) - Range of \( h(x) \): \( (-\infty, \infty) \) - Domain of \( h^{-1}(y) \): \( (-\infty, \infty) \) - Range of \( h^{-1}(y) \): \( (-\infty, \infty) \) --- ### d) \( p(x) = 3x^{2} \) **i) Determine the equation of the inverse:** To find the inverse, we set \( y = 3x^{2} \) and solve for \( x \): \[ y = 3x^{2} \implies x = \pm \frac{\sqrt{3y}}{3} \] Since \( p(x) \) is an upward-opening parabola, we restrict the domain to \( x \geq 0 \) to ensure the inverse is a function: \[ p^{-1}(y) = \frac{\sqrt{3y}}{3} \] **ii) Graph the relation, its inverse, and the line \( y=x \):** - The graph of \( p(x) \) is an upward parabola. - The inverse \( p^{-1}(y) \) is the right half of the upward parabola. - The line \( y=x \) is a diagonal line through the origin. **iii) Domain and Range:** - Domain of \( p(x) \): \( [0, \infty) \) - Range of \( p(x) \): \( [0, \infty) \) - Domain of \( p^{-1}(y) \): \( [0, \infty) \) - Range of \( p^{-1}(y) \): \( [0, \infty) \) --- This concludes the analysis of the functions and their inverses. 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