Question

PRE-CAL
show that the given function is one-to-one and find its inverse. Check your answers algebraically and graphically. Verify that the range of f is the domain of f^-1(x)

f(x)= 3/4-

Answer 1

is it
\[f(x)=\frac{3}{4-x}\]?

Answer 2

yes
@satellite73

Answer 3

there are a couple of ways to show that it is one to one
one way is to graph it and see that it passes the horizontal line test
https://www.wolframalpha.com/input/?i=3%2F%284-x%29

Answer 4

another way is to put
\[f(a)=f(b)\] and show via algebra that \(a=b\)
that is, start with
\[\frac{3}{4-a}=\frac{3}{4-b}\]

Answer 5

you get
\[3(4-b)=3(4-a)\iff12-3b=12-3a\]
\[\iff-3b=3a\iff b=a\]

Answer 6

another way is to simply find the inverse
write \(y=\frac{3}{4-x}\) switch \](x\) and \(y\) because that is what the inverse does, get
\[x=\frac{3}{4-y}\] and solve for \(y\)

Answer 7

\[x(4-y)=3\]
\[4x-xy=3\]\[-xy=-3-4x\]\[y=\frac{-3-4x}{-x}\] or
\[y=\frac{4x+3}{x}\]

Answer 8

the domain of the inverse is all numbers except zero, which is also the range of the original function

Answer 9

thank you you have really helped me thanks!!!