LET F(X)=7X+1 AND LET G(X)= X-1/7, COMPUTE F o g(x) and g o f(x). what do your answer mean?

LET F(X)=7X+1 AND LET G(X)= X-1/7, COMPUTE F o g(x) and g o f(x). what do your answer mean?

another problem is
use the one to one property to solve the equation for x
1.. 3^x+1=27
2...... 2^x-3=16
3... 2^x-2=1/32
4.... (1/5)^x+1=125
5... e^2x-1= e4
6... e^x2+6=e^5x

please help any steps to follow or easy way to learn those exercise or any video

1 Answer
Sean
Apr 29, 2018

Please see below.

Explanation:

.

You need to enclose the #(x-1)# in #g(x)# in parentheses. The same goes for all the exponents in problems #1.# through #6.#. Otherwise, you would have completely different problems.

#f(x)=7x+1#

#g(x)=(x-1)/7#

#f@g(x)=f(g(x))=7((x-1)/7)+1=x-1+1=x#

#g@f(x)=g(f(x))=(7x+1-1)/7=(7x)/7=x#

This means that #f(x)# and #g(x)# are inverse functions of each other.

Using one to one property, we solve the following:

#color(red)1.# #color(red)(3^(x+1)=27)#

#3^(x+1)=3^3#

One to one property means the only way this can be true is if:

#x+1=3#

#x=2#

#color(red)2.# #color(red)(2^(x-3)=16)#

#2^(x-3)=2^4#

#x-3=4#

#x=7#

#color(red)3.# #color(red)(2^(x-2)=1/32)#

#2^(x-2)=1/2^5=2^(-5)#

#x-2=-5#

#x=-3#

#color(red)4.# #color(red)((1/5)^(x+1)=125)#

#(5^(-1))^(x+1)=5^3#

#5^(-x-1)=5^3#

#-x-1=3#

#x=-4#

#color(red)5.# #color(red)(e^(2x-1)=e^4)#

#2x-1=4#

#2x=5#

#x=5/2#

#color(red)6.# #color(red)(e^(x^2+6)=e^(5x))#

#x^2+6=5x#

#x^2-5x+6=0#

#(x-6)(x+1)=0#

#x=6 and -1#

Here are some videos to watch to learn how to solve these types of problems:

How to solve exponential problems using one to one property

How to solve exponential problems using one to one property

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