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The function [tex]$g$[/tex] is defined by [tex]$g(x)=\frac{1}{3x-7}$[/tex], and [tex][tex]$g(b)=20$[/tex][/tex], where [tex]$b$[/tex] is a constant. What is the value of [tex]$b$[/tex]?

(A) [tex]$\frac{1}{23}$[/tex]
(B) [tex][tex]$\frac{141}{60}$[/tex][/tex]
(C) 9
(D) 81


Sagot :

To determine the value of [tex]\(b\)[/tex] given the function [tex]\( g(x) = \frac{1}{3x - 7} \)[/tex] and the equation [tex]\( g(b) = 20 \)[/tex], we proceed as follows:

1. Start by setting up the equation:
[tex]\[ g(b) = 20 \][/tex]
Substituting the definition of [tex]\( g(x) \)[/tex] into this equation gives:
[tex]\[ \frac{1}{3b - 7} = 20 \][/tex]

2. Solve for [tex]\( b \)[/tex]:

First, take the reciprocal of both sides to get rid of the fraction:
[tex]\[ 3b - 7 = \frac{1}{20} \][/tex]

3. Isolate [tex]\( b \)[/tex]:

Add 7 to both sides of the equation:
[tex]\[ 3b = \frac{1}{20} + 7 \][/tex]

To combine terms, rewrite [tex]\( 7 \)[/tex] as a fraction with a common denominator:
[tex]\[ 7 = \frac{140}{20} \][/tex]
Then add:
[tex]\[ 3b = \frac{1}{20} + \frac{140}{20} = \frac{141}{20} \][/tex]

4. Solve for [tex]\( b \)[/tex] by dividing both sides by 3:
[tex]\[ b = \frac{\frac{141}{20}}{3} = \frac{141}{20} \times \frac{1}{3} = \frac{141}{60} \][/tex]

Thus, the value of [tex]\( b \)[/tex] is:
[tex]\[ \boxed{\frac{141}{60}} \][/tex]

Therefore, the correct answer is (B) [tex]\( \frac{141}{60} \)[/tex].