Answer :
To determine the domain and range of the function [tex]\( g(x) = \frac{-3}{x - 6} + 2 \)[/tex], let's analyze the function step by step.
### Domain:
The domain of a function consists of all the possible input values (x-values) for which the function is defined.
1. The function [tex]\( g(x) = \frac{-3}{x - 6} + 2 \)[/tex] includes a fraction with a denominator of [tex]\( x - 6 \)[/tex].
2. The value of the denominator must not be zero because division by zero is undefined.
3. To find the value of [tex]\( x \)[/tex] that causes the denominator to be zero, we solve:
[tex]\[ x - 6 = 0 \][/tex]
[tex]\[ x = 6 \][/tex]
4. Since [tex]\( x = 6 \)[/tex] makes the denominator zero and the function undefined at this point, we must exclude [tex]\( x = 6 \)[/tex] from the domain.
Thus, the domain of [tex]\( g(x) \)[/tex] is all real numbers except [tex]\( x = 6 \)[/tex].
### Range:
The range of a function consists of all the possible output values (y-values) that the function can produce.
1. Consider the function in the form [tex]\( g(x) = \frac{-3}{x - 6} + 2 \)[/tex]. Let's isolate the term with the variable:
[tex]\[ g(x) - 2 = \frac{-3}{x - 6} \][/tex]
2. To understand the behavior of [tex]\( g(x) \)[/tex], we examine the properties of the term [tex]\( \frac{-3}{x - 6} \)[/tex].
3. As [tex]\( x \)[/tex] approaches 6 from either side, [tex]\( \frac{-3}{x - 6} \)[/tex] approaches [tex]\( \pm \infty \)[/tex], depending on the direction from which [tex]\( x \)[/tex] approaches 6.
4. Consequently, [tex]\( g(x) \)[/tex] will also approach [tex]\( \pm \infty \)[/tex], but it will never actually reach a value where [tex]\( \frac{-3}{x - 6} = 0 \)[/tex]. For [tex]\( \frac{-3}{x - 6} = 0 \)[/tex], [tex]\( g(x) \)[/tex] would need to be 2.
5. Therefore, the function [tex]\( g(x) \)[/tex] cannot equal 2 because that would imply [tex]\( \frac{-3}{x - 6} = 0 \)[/tex], which is impossible.
Thus the range of [tex]\( g(x) \)[/tex] is all real numbers except 2.
### Conclusion:
The domain is all real numbers except [tex]\( 6 \)[/tex]:
[tex]\[ \text{Domain: } (-\infty, 6) \cup (6, \infty) \][/tex]
The range is all real numbers except [tex]\( 2 \)[/tex]:
[tex]\[ \text{Range: } (-\infty, 2) \cup (2, \infty) \][/tex]
So, filling in the blanks:
The domain is all real numbers except [tex]\( 6 \)[/tex] and the range is all real numbers except [tex]\( 2 \)[/tex].
### Domain:
The domain of a function consists of all the possible input values (x-values) for which the function is defined.
1. The function [tex]\( g(x) = \frac{-3}{x - 6} + 2 \)[/tex] includes a fraction with a denominator of [tex]\( x - 6 \)[/tex].
2. The value of the denominator must not be zero because division by zero is undefined.
3. To find the value of [tex]\( x \)[/tex] that causes the denominator to be zero, we solve:
[tex]\[ x - 6 = 0 \][/tex]
[tex]\[ x = 6 \][/tex]
4. Since [tex]\( x = 6 \)[/tex] makes the denominator zero and the function undefined at this point, we must exclude [tex]\( x = 6 \)[/tex] from the domain.
Thus, the domain of [tex]\( g(x) \)[/tex] is all real numbers except [tex]\( x = 6 \)[/tex].
### Range:
The range of a function consists of all the possible output values (y-values) that the function can produce.
1. Consider the function in the form [tex]\( g(x) = \frac{-3}{x - 6} + 2 \)[/tex]. Let's isolate the term with the variable:
[tex]\[ g(x) - 2 = \frac{-3}{x - 6} \][/tex]
2. To understand the behavior of [tex]\( g(x) \)[/tex], we examine the properties of the term [tex]\( \frac{-3}{x - 6} \)[/tex].
3. As [tex]\( x \)[/tex] approaches 6 from either side, [tex]\( \frac{-3}{x - 6} \)[/tex] approaches [tex]\( \pm \infty \)[/tex], depending on the direction from which [tex]\( x \)[/tex] approaches 6.
4. Consequently, [tex]\( g(x) \)[/tex] will also approach [tex]\( \pm \infty \)[/tex], but it will never actually reach a value where [tex]\( \frac{-3}{x - 6} = 0 \)[/tex]. For [tex]\( \frac{-3}{x - 6} = 0 \)[/tex], [tex]\( g(x) \)[/tex] would need to be 2.
5. Therefore, the function [tex]\( g(x) \)[/tex] cannot equal 2 because that would imply [tex]\( \frac{-3}{x - 6} = 0 \)[/tex], which is impossible.
Thus the range of [tex]\( g(x) \)[/tex] is all real numbers except 2.
### Conclusion:
The domain is all real numbers except [tex]\( 6 \)[/tex]:
[tex]\[ \text{Domain: } (-\infty, 6) \cup (6, \infty) \][/tex]
The range is all real numbers except [tex]\( 2 \)[/tex]:
[tex]\[ \text{Range: } (-\infty, 2) \cup (2, \infty) \][/tex]
So, filling in the blanks:
The domain is all real numbers except [tex]\( 6 \)[/tex] and the range is all real numbers except [tex]\( 2 \)[/tex].
