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Solve for \( x \):
[tex]\[ 3x = 6x - 2 \][/tex]



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$
f(x)=\frac{a-19}{x}+5
$

In the given function \( f, a \) is a constant. The graph of function \( f \) in the \( x y \)-plane, where \( y=f(x) \), is translated 3 units down and 4 units to the right to produce the graph of \( y=g(x) \). Which equation defines function \( g \) ?
(A) \( g(x)=\frac{a-19}{x+4}+2 \)
(B) \( g(x)=\frac{a-19}{x-4}+2 \)
(C) \( g(x)=\frac{a-22}{x+4}+5 \)
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Response:
Given the function \( f(x)=\frac{a-19}{x}+5 \), where \( a \) is a constant, the graph of function \( f \) in the \( xy \)-plane, where \( y=f(x) \), is translated 3 units down and 4 units to the right to produce the graph of \( y=g(x) \).

Which equation defines function \( g \)?

(A) \( g(x)=\frac{a-19}{x+4}+2 \)

(B) \( g(x)=\frac{a-19}{x-4}+2 \)

(C) [tex]\( g(x)=\frac{a-22}{x+4}+5 \)[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to determine how the given function \( f(x) = \frac{a-19}{x} + 5 \) transforms when the graph is translated 3 units down and 4 units to the right to produce the graph of \( y = g(x) \).

### Step-by-Step Solution:

1. Translation Basics:
- Translating a graph to the right by \( h \) units implies substituting \( x \) with \( (x - h) \) in the function.
- Translating a graph downward by \( k \) units means subtracting \( k \) from the entire function.

2. Translate 4 Units to the Right:
- To move the graph 4 units to the right, we substitute \( x \) with \((x - 4)\) in \( f(x) \):

[tex]\[ f(x - 4) = \frac{a-19}{x-4} + 5 \][/tex]

3. Translate 3 Units Down:
- To move the graph 3 units down, we subtract 3 from the result of the previous step:

[tex]\[ g(x) = \left(\frac{a-19}{x-4} + 5\right) - 3 = \frac{a-19}{x-4} + (5 - 3) \][/tex]

4. Simplify the Translated Function:
- Simplify the expression within the parentheses:

[tex]\[ g(x) = \frac{a-19}{x-4} + 2 \][/tex]

This matches option (B):

[tex]\[ g(x) = \frac{a-19}{x-4} + 2 \][/tex]

5. Verify Against Other Options:
- Option (A): \( g(x) = \frac{a-19}{x+4} + 2 \)
- This is incorrect because the \( x \) in the denominator should be \( (x - 4) \), not \( (x + 4) \).
- Option (C): \( g(x) = \frac{a-22}{x+4} + 5 \)
- This is also incorrect because neither the numerator \((a-22)\) nor the \( x \) term in the denominator \((x+4)\) matches the required transformations.

Therefore, the correct equation defining the function \( g(x) \) is:

[tex]\[ \boxed{B} \][/tex]

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