Answer to Question #149167 in Algebra for Patrick J

Question

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 1

$f(y)=\frac{4}{y-2}, g(x)=\frac{5}{3x-1}$

$f\circ g=\frac{4(3x-1)}{7-6x}$

We have:

for domain of $g(x)$ : $x\neq1/3$

for domain of $f(y)$ : $y\neq2$ $\implies \frac{5}{3x-1}\neq2\implies x\neq7/6$

for domain of $f\circ g$ : $x\neq7/6$

So, resulting domain of $f\circ g$ : $x\isin(-\infin,1/3)\bigcup(1/3,7/6)\bigcup(7/6, \infin)$

$f(x)=4+\sqrt{x-2}$ , domain: $x\isin[2,\infin)$ ; range: $y\isin[4,\infin)$

$x=g(y)=(y-4)^2+2,\, y \geq 4.$

Changing variables x and y we get inverse function:

$f^{-1}(x)=(x-4)^2+2$ , domain: $x\isin[4,\infin)$, range: $y\isin[2,\infin)$

Points of the form (a,b),(b,a) are reflected about the midpoint between the two points.

In our case the midpoint between (11,7) and (7,11) is (9,9).

So, (11,7) and (7,11) are reflected about (9,9).