Find the domain and range for the indicated function.
-$f ( x ) = \sqrt { x + 13 } , \quad g ( x ) = \sqrt { x - 13 }$ ; $g-f$
A) D: $x \geq 13$
R: $y \geq 0$

B) $\mathrm { D } : \mathrm { x } \geq 13$
R: $y \geq - \sqrt { 26 }$

C) D: $x \geq 13$
R: $y \geq \sqrt { 26 }$

D) D: $x \geq - 13$
R: $y \geq - \sqrt { 26 }$

Question

Find the domain and range for the indicated function.
-$f ( x ) = \sqrt { x + 13 } , \quad g ( x ) = \sqrt { x - 13 }$ ; $g-f$
A) D: $x \geq 13$
R: $y \geq 0$

B) $\mathrm { D } : \mathrm { x } \geq 13$
R: $y \geq - \sqrt { 26 }$

C) D: $x \geq 13$
R: $y \geq \sqrt { 26 }$

D) D: $x \geq - 13$
R: $y \geq - \sqrt { 26 }$

Quizplus · Accepted Answer

To find the domain of g-f, we need to first find the domain of both g and f. Since both g and f involve taking the square root of a quantity, we want to make sure that the quantity inside the square root is nonnegative. Therefore, we have:

Domain of f: 
x + 13 ≥ 0 
x ≥ -13

Domain of g: 
x - 13 ≥ 0 
x ≥ 13

So the domain of g-f is the intersection of these two domains, which is:

x ≥ 13

To find the range of g-f, we will use the fact that the range of g-f is the set of all possible differences between elements of the range of g and elements of the range of f. We have:

Range of f: 
y = √(x+13) 
y ≥ 0 (since the square root of a nonnegative quantity is always nonnegative)

Range of g: 
y = √(x-13) 
y ≥ 0 (since the square root of a nonnegative quantity is always nonnegative)

So the range of g-f is given by the set of all differences between nonnegative numbers. The smallest possible difference is when g is smallest and f is largest, which occurs when x equals 13. In this case, g(13) = 0 and f(13) = √26, so g-f(13) = -√26. The largest possible difference is when g is largest and f is smallest, which occurs when x approaches infinity. In this case, both g and f approach infinity, so the difference between them approaches zero.

Therefore, the range of g-f is y ≥ -√26.

Find the Domain and Range for the Indicated Function f(x)=x+13,g(x)=x−13f ( x ) = \sqrt { x + 13 } , \quad g ( x ) = \sqrt { x - 13 }f(x)=x+13​,g(x)=x−13​

Find the Domain and Range for the Indicated Function $f ( x ) = \sqrt { x + 13 } , \quad g ( x ) = \sqrt { x - 13 }$