SOLUTION: What can be said about the domain of the function f \circ g where f(y)= \frac{4}{y-2} and g(x)= \frac{5}{3x-1} ? Express it in terms of a union of intervals of real numbers. G

Question 1171467: What can be said about the domain of the function f \circ g where f(y)= \frac{4}{y-2} and g(x)= \frac{5}{3x-1} ? Express it in terms of a union of intervals of real numbers. Go to www.desmos.com/calculator and obtain the graph of f , g , and f \circ g .
Find the inverse of the function f(x)=4+ \sqrt{x-2} .
State the domains and ranges of both the function and the inverse function in terms of intervals of real numbers.
Go to www.desmos.com/calculator and obtain the graph of f , its inverse, and g(x)=x in the same system of axes. About what pair (a, a) are (11, 7) and (7, 11) reflected about?
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
This question is identical to a previous question. Here's the solution again:
**1. Domain of f(g(x)) (f \circ g)**
* **f(y) = 4/(y - 2)**
* The domain of f(y) is all real numbers except y = 2.
* **g(x) = 5/(3x - 1)**
* The domain of g(x) is all real numbers except 3x - 1 = 0, which means x = 1/3.
* **f(g(x)) = f(5/(3x - 1)) = 4 / [(5/(3x - 1)) - 2]**
* To find the domain of f(g(x)), we need to consider two things:
* The domain of g(x) (x ≠ 1/3).
* The values of x that make the denominator of f(g(x)) equal to zero.
* Set the denominator to zero:
* 5/(3x - 1) - 2 = 0
* 5/(3x - 1) = 2
* 5 = 2(3x - 1)
* 5 = 6x - 2
* 7 = 6x
* x = 7/6
* Therefore, the domain of f(g(x)) is all real numbers except x = 1/3 and x = 7/6.
* In interval notation: (-∞, 1/3) U (1/3, 7/6) U (7/6, ∞)
* **Desmos Graphing:**
* Go to [www.desmos.com/calculator](https://www.desmos.com/calculator).
* Input:
* f(y) = 4/(y - 2)
* g(x) = 5/(3x - 1)
* f(g(x))
* Examine the graph of f(g(x)) to confirm the domain. You will see vertical asymptotes at x=1/3 and x=7/6.
**2. Inverse of f(x) = 4 + √(x - 2)**
* **Finding the inverse:**
* Let y = 4 + √(x - 2)
* Swap x and y: x = 4 + √(y - 2)
* Solve for y:
* x - 4 = √(y - 2)
* (x - 4)² = y - 2
* y = (x - 4)² + 2
* Therefore, f⁻¹(x) = (x - 4)² + 2
* **Domains and Ranges:**
* **f(x) = 4 + √(x - 2)**
* Domain: x - 2 ≥ 0 => x ≥ 2, or [2, ∞)
* Range: Since √(x - 2) ≥ 0, f(x) ≥ 4, or [4, ∞)
* **f⁻¹(x) = (x - 4)² + 2**
* Domain: The range of f(x) becomes the domain of f⁻¹(x), so x ≥ 4, or [4, ∞)
* Range: The domain of f(x) becomes the range of f⁻¹(x), so y ≥ 2, or [2, ∞)
* **Desmos Graphing:**
* Go to [www.desmos.com/calculator](https://www.desmos.com/calculator).
* Input:
* f(x) = 4 + sqrt(x - 2)
* y = (x - 4)² + 2 {x>=4}
* y = x
* The graph of f⁻¹(x) is the reflection of f(x) across the line y = x.
* The points (11, 7) and (7, 11) are reflections of each other across the line y = x.
* The point that is the center of the reflection is the intersection of the line y=x, and the line that goes through (11,7) and (7,11). The midpoint of (7,11) and (11,7) is (9,9). Therefore, the points are reflected about the pair (9, 9).
RELATED QUESTIONS
Express the function h(x)=\frac{1}{x + 9} in the form f \circ g. If g(x)=x + 9, find the... (answered by jim_thompson5910)

Find the all real numbers that are not in the domain of f(g(x)), where f(x) = \frac{3x^2 (answered by CPhill)

What can be said about the domain of the function f . g where f(y)= {4}/{y-2} and... (answered by CPhill)

If \[\frac{x}{y} = \frac{4}{5}, \; \frac{y}{z} = \frac{2}{5}, \;\text{and} \; \frac{z}{w} (answered by ikleyn,josgarithmetic,greenestamps,math_tutor2020)

Find the largest value of x where the plots of f(x) = - \frac{2x + 5}{x + 3} and g(x) =... (answered by CPhill,ikleyn,greenestamps,Edwin McCravy)

Find the values of x where the vertical asymptotes of f(g(x)) are located, where f(x) =... (answered by CPhill)

What happens if we graph both f and f^{-1} on the same set of axes, using the x-axis... (answered by Theo)

Give a polynomial g(x) so that f(x) + g(x) has a horizontal asymptote of y = 0 as x... (answered by CPhill)

Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 +... (answered by math_tutor2020,ikleyn)