Let g(x)=(x-5)/(sqrt(x-4)-1) when x!=5. g is continuous for all x > 4. Find g(5). Choose 1 answer: (A) 2 (B) 8 (c) 10 (D) 5

Given function: We are given the function g(x)=(x-5)/(sqrt(x-4)-1) and we need to find the value of g(5). However, we cannot directly substitute x=5 into the function because it would result in a division by zero, which is undefined. We need to find a way to simplify the function or use a limit approach to find the value of g(5).Simplifying the function: To simplify the function, we can multiply the numerator and the denominator by the conjugate of the denominator to eliminate the square root. The conjugate of sqrt(x-4)-1 is sqrt(x-4)+1.Multiplying by the conjugate: Multiplying the numerator and denominator by the conjugate, we get: g(x) = [(x-5)(sqrt(x-4)+1)] / [(sqrt(x-4)-1)(sqrt(x-4)+1)]Simplifying the denominator: Simplifying the denominator using the difference of squares, we get: g(x) = [(x-5)(sqrt(x-4)+1)] / [(x-4) - 1]Canceling out terms: Further simplifying the denominator, we get: g(x) = [(x-5)(sqrt(x-4)+1)] / [x - 5]Substituting x=5: Now we can see that the (x-5) terms in the numerator and denominator will cancel out, as long as x is not equal to 5. This gives us: g(x) = sqrt(x-4) + 1Calculating the result: Since we have simplified the function and there is no longer a division by zero, we can now substitute x=5 into the simplified function to find g(5): g(5) = sqrt(5-4) + 1Calculating the square root and the addition, we get: g(5) = sqrt(1) + 1 = 1 + 1 = 2

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Let
g(x)=(x-5)/(sqrt(x-4)-1) when
x!=5.

g is continuous for all
x > 4.
Find
g(5).
Choose 1 answer:
(A) 2
(B) 8
(c) 10
(D) 5

Let $g(x)=\frac{x-5}{\sqrt{x-4}-1}$ when $x \neq 5$ . $\newline$ $g$ is continuous for all x>4 . $\newline$ Find $g(5)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $2$ $\newline$ (B) $8$ $\newline$ (C) $10$ $\newline$ (D) $5$

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Math Problems

Algebra 1

Domain and range of square root functions: equations

Full solution

Q. Let

g(x)=\frac{x-5}{\sqrt{x-4}-1}

when

x \neq 5

\newline

g

is continuous for all

x>4

\newline

Find

g(5)

\newline

Choose

1

answer:

\newline

(A)

2

\newline

(B)

8

\newline

(C)

10

\newline

(D)

5

Given function: We are given the function $g(x)=\frac{x-5}{\sqrt{x-4}-1}$ and we need to find the value of $g(5)$ . However, we cannot directly substitute $x=5$ into the function because it would result in a division by zero, which is undefined. We need to find a way to simplify the function or use a limit approach to find the value of $g(5)$ .
Simplifying the function: To simplify the function, we can multiply the numerator and the denominator by the conjugate of the denominator to eliminate the square root. The conjugate of $\sqrt{x-4}-1$ is $\sqrt{x-4}+1$ .
Multiplying by the conjugate: Multiplying the numerator and denominator by the conjugate, we get: $\newline$ g(x) = $\frac{(x-5)(\sqrt{x-4}+1)}{(\sqrt{x-4}-1)(\sqrt{x-4}+1)}$
Simplifying the denominator: Simplifying the denominator using the difference of squares, we get: $g(x) = \frac{(x-5)(\sqrt{x-4}+1)}{(x-4) - 1}$
Canceling out terms: Further simplifying the denominator, we get: $\newline$ g(x) = $[\$x-5$ $\sqrt{x-4}+1$ ] / [x - $5$ ]\)
Substituting $x=5$ : Now we can see that the $(x-5)$ terms in the numerator and denominator will cancel out, as long as $x \neq 5$ . This gives us: $\newline$ $g(x) = \sqrt{x-4} + 1$
Calculating the result: Since we have simplified the function and there is no longer a division by zero, we can now substitute $x=5$ into the simplified function to find $g(5)$ : $g(5) = \sqrt{5-4} + 1$
Calculating the result: Since we have simplified the function and there is no longer a division by zero, we can now substitute $x=5$ into the simplified function to find $g(5)$ :
$g(5) = \sqrt{5-4} + 1$ Calculating the square root and the addition, we get:
$g(5) = \sqrt{1} + 1 = 1 + 1 = 2$