Question 1179038: Graph f as a solid line and F-1 as a dashed line in the same rectangular coordinate space. Use interval notation to give the domain and range of f and F-1.
f(x) = 3x - 2
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to graph f and f⁻¹ and find their domain and range:
**1. Graph f(x) = 3x - 2:**
* This is a linear function with a slope of 3 and a y-intercept of -2.
* Plot the y-intercept (0, -2).
* Use the slope to find another point: from (0, -2), move up 3 units and right 1 unit to get (1, 1).
* Draw a solid line through these points.
**2. Find f⁻¹(x):**
* Replace f(x) with y: y = 3x - 2
* Swap x and y: x = 3y - 2
* Solve for y:
* x + 2 = 3y
* y = (x + 2) / 3
* Replace y with f⁻¹(x): f⁻¹(x) = (x + 2) / 3
**3. Graph f⁻¹(x) = (x + 2) / 3:**
* This is also a linear function with a slope of 1/3 and a y-intercept of 2/3.
* Plot the y-intercept (0, 2/3).
* Use the slope to find another point: from (0, 2/3), move up 1 unit and right 3 units to get (3, 5/3).
* Draw a dashed line through these points.
**4. Domain and Range:**
* **f(x) = 3x - 2:**
* Domain: All real numbers (-∞, ∞)
* Range: All real numbers (-∞, ∞)
* **f⁻¹(x) = (x + 2) / 3:**
* Domain: All real numbers (-∞, ∞)
* Range: All real numbers (-∞, ∞)
**Note:**
* The graphs of f and f⁻¹ are reflections of each other across the line y = x.
* Since both functions are linear, their domains and ranges are all real numbers.
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