Question

Consider the functions fix)#2x + 7 and g(x)-2(x-7) (a) Find f(g/x)). (b) Find g(fx)) of each other O B. No values should be excluded from the domain (b) What is gtt(x)? Give any values of x that need to be excluded from g/f(x). Select the correct choice below and fil in any answer boxes within your choice (Use a comma to separate answers as needed) O B. No values should be excluded from the domair (c) Are the functions f and g inverses of each other? Choose the corret answer below O No O Yes

Answer 1

Answer: (a) f(g(x))=x , (b) g(f(x))=x and (c) both f(x) and g(x) are inverse of each other.

Explanation:

(a) f(x)=2x+7 and g(x)=(1/2)(x-7).

$f(g(x))=f(\frac{1}{2}(x-7))$

$=2\times \left (\frac{1}{2}(x-7) \right )+7$

$=x-7+7$

$=x$

so, f(g(x))=x

(b) g(f(x))=?

$g(f(x))=g(2x+7)$

$=\frac{1}{2}(2x+7-7)$

$=x$

So, g(f(x))=x

(c) For determination if the function f(x) and g(x) are inverse of each other, f^-1(x)=g(x) and g^-1(x)=f(x)

For f^-1(x)

f(x)=y=2x+7

y-7=2x

$\frac{1}{2}(y-7)=x$

interchangig y and x.

$\frac{1}{2}(x-7)=y=f^-^1(x)$

$So\Rightarrow f^-^1(x)=\frac{1}{2}(x-7)$ ----------------------------------------------------- (1)

Now, for g^-1(x)

$g(x)=y=\frac{1}{2}(x-7)$

$2y=(x-7)$

$2y+7=x$

interchanig y and x.

$2x+7=y=g^-^1(x)$

$So\Rightarrow g^-^1(x)=2x+7$ ------------------------------------------------- (2)

So, we observe from eqn 1 and 2 that, f^-1(x)=(1/2)(x-7)=g(x). and g^-1(x)=2x+7=f(x).

Hence, both f(x) and g(x) are inverse of each other.

(B) No values should be excluded from the domain of g(f(x)).