Question

Find the inverse of each function.

a. f(x) = 3x b. f(x) = (1/2)^x
c. g(x) = ln(x − 7) d. h(x) = log3(x + 2) / log3(5)
e. f(x) = 3(1.8)0.2x + 3 f. g(x) = log2(3√x − 4)
g. h(x) = 5x / 5x +1
h. f(x) = 2−x+1

Answer 1

a. f(x)^(-1)=x/3

b. f(x)^(-1)=log_10⁡x/log_10⁡〖1/2〗

c. g(x)^(-1)=e^x+7

d. f(x)^(-1)=5^x-2

e. f(x)^(-1)=(x-3)/1.08

f. f(x)^(-1)=((2^x-4)/3)^2

g. h(x)^(-1)=x/(5-5x)

h. f(x)^(-1)=x+3

Explanation:

To understand how to find the inverse function you must solve the equation for x and then replace x by the definition of the inverse f(x)-1. Then the explanation of each exercise goes as follows:

a. f(x)=3x

f(x)/3=x

f(x)^(-1)=x/3

b.f(x)=(1/2)^(x )

Considering that log_b⁡(a)=c and log_b⁡(a)=log_c⁡(a)/log_c⁡(b)

log_(1/2) f(x)=log_(1/2)⁡((1/2)^x )

log_(1/2) f(x)=x

f(x)^(-1)=log_(1/2) (x)=log_10⁡(x)/log_10⁡(1/2)

c. g(x)=ln⁡(x-7)

Considering that ln⁡(e^x )=x

e^g(x) =e^ln⁡(x-7)

e^g(x) =(x-7)

x=e^g(x) +7

g(x)^(-1)=e^x+7

d. h(x)=log_3⁡(x+2)/log_3⁡(5)

log_3⁡(x+2)/log_3⁡(5) =log_5⁡(x+2)

h(x)=log_5⁡(x+2)

5^h(x) =x+2

h(x)^(-1)=5^x-2

e. f(x)=3*1.8*0.2*x+3

f(x)=3*1.8*0.2*x+3

f(x)^(-1)=(x-3)/1.08

f. f(x)=log_2⁡(3√x-4)

2^f(x) =2^log_2⁡(3√x-4)

2^f(x) -4=3√x

((2^f(x) -4)/3)^2=√x^2

f(x)^(-1)=((2^x-4)/3)^2

g. h(x)=5x/(5x+1)

(5x+1)h(x)=5x

h(x)=5x(1-h(x))

x=h(x)/5(1-h(x))

h(x)^(-1)=x/(5(1-x))

h. f(x)=2-x+1=3-x

f(x)^(-1)=3+x