Responder
a) \( f(x) = -\frac{3}{2} x^{2} \):
- Inverse: \( f^{-1}(y) = -\frac{\sqrt{-6y}}{3} \)
- Graph: Downward parabola, its inverse is the left half of an upward parabola, and the line \( y=x \) is a diagonal line.
- Domain and Range: Both \( f(x) \) and \( f^{-1}(y) \) have domains and ranges of \( (-\infty, 0] \).
b) \( g(x) = -\frac{3}{2} x \):
- Inverse: \( g^{-1}(y) = -\frac{2}{3} y \)
- Graph: Straight lines with negative slopes, and the line \( y=x \) is a diagonal line.
- Domain and Range: Both \( g(x) \) and \( g^{-1}(y) \) have domains and ranges of \( (-\infty, \infty) \).
c) \( h(x) = 3x \):
- Inverse: \( h^{-1}(y) = \frac{y}{3} \)
- Graph: Straight lines with positive slopes, and the line \( y=x \) is a diagonal line.
- Domain and Range: Both \( h(x) \) and \( h^{-1}(y) \) have domains and ranges of \( (-\infty, \infty) \).
d) \( p(x) = 3x^{2} \):
- Inverse: \( p^{-1}(y) = \frac{\sqrt{3y}}{3} \)
- Graph: Upward parabola, its inverse is the right half of an upward parabola, and the line \( y=x \) is a diagonal line.
- Domain and Range: Both \( p(x) \) and \( p^{-1}(y) \) have domains and ranges of \( [0, \infty) \).
Solución
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. If you need any further assistance or visualizations, please let me know!
Revisado y aprobado por el equipo de tutoría de UpStudy
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