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What are the domain and range of the function [tex]\( f(x) = -\log(5 - x) + 9 \)[/tex]?

A. Domain: [tex]\( x \ \textless \ 5 \)[/tex], Range: [tex]\( y \geq 9 \)[/tex]

B. Domain: [tex]\( x \ \textless \ 5 \)[/tex], Range: [tex]\( (-\infty, \infty) \)[/tex]

C. Domain: [tex]\( x \geq 9 \)[/tex], Range: [tex]\( (-\infty, \infty) \)[/tex]

D. Domain: [tex]\( x \geq 9 \)[/tex], Range: [tex]\( y \geq 9 \)[/tex]

Sagot :

GinnyAnswer
To determine the domain and range of the function [tex]\( f(x) = -\log(5-x) + 9 \)[/tex], let’s analyze it step by step.

### 1. Domain
The domain of the function is determined by the values of [tex]\( x \)[/tex] for which the function is defined.

The logarithmic function [tex]\(\log(5-x)\)[/tex] is defined when its argument [tex]\( (5-x) \)[/tex] is positive, i.e., [tex]\( 5 - x > 0 \)[/tex].

Solving this inequality:
[tex]\[ 5 - x > 0 \\ 5 > x \\ x < 5 \][/tex]

So, the domain of [tex]\( f(x) \)[/tex] is [tex]\( x < 5 \)[/tex].

### 2. Range
To find the range, we need to consider the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches the boundary of its domain and within its domain.

#### Behavior as [tex]\( x \)[/tex] approaches 5 from the left:
As [tex]\( x \)[/tex] approaches 5 from the left ([tex]\( x \to 5^- \)[/tex]), [tex]\( 5 - x \)[/tex] approaches 0 from the positive side. Since [tex]\(\log(0^+) \to -\infty\)[/tex],

[tex]\[ \log(5 - x) \to -\infty \][/tex]

Hence,
[tex]\[ -\log(5 - x) \to \infty \][/tex]

Adding 9 to this,
[tex]\[ f(x) \to \infty \][/tex]

#### Behavior within the domain ([tex]\( x < 5 \)[/tex]):
As [tex]\( x \)[/tex] decreases, for example, when [tex]\( x \)[/tex] approaches [tex]\(-\infty \)[/tex] within the allowed domain which is [tex]\( x < 5 \)[/tex], the value of [tex]\(\log(5 - x)\)[/tex] becomes very large positively since [tex]\( 5 - x \)[/tex] becomes large. Thus:

[tex]\[ \log(5 - x) \to \infty \][/tex]

[tex]\[ -\log(5 - x) \to -\infty \][/tex]

Adding 9 to this,
[tex]\[ f(x) \to -\infty \][/tex]

Putting these observations together, we see that as [tex]\(x\)[/tex] varies over its domain ([tex]\(-\infty < x < 5\)[/tex]), the function [tex]\( f(x) \)[/tex] covers all values from [tex]\(-\infty\)[/tex] to [tex]\(+\infty\)[/tex].

Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].

### Conclusion
The solution to the question is:

- Domain: [tex]\( x < 5 \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]

Thus, the correct answer is:

[tex]\( \text{domain: } x < 5 \text{ range: } (-\infty, \infty) \)[/tex].
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