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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 the function step-by-step:

### Domain
1. The function involves a logarithmic term [tex]\(\log(5-x)\)[/tex]. The logarithmic function [tex]\(\log(z)\)[/tex] is defined only for positive arguments [tex]\(z > 0\)[/tex].
2. Hence, the argument of the logarithm, [tex]\(5 - x\)[/tex], must be positive:
[tex]\[ 5 - x > 0 \][/tex]
3. Solving this inequality for [tex]\(x\)[/tex], we get:
[tex]\[ x < 5 \][/tex]
So, the domain of the function [tex]\(f(x)\)[/tex] is all [tex]\(x\)[/tex] such that [tex]\(x < 5\)[/tex].

### Range
1. To determine the range, consider the behavior of the function [tex]\(f(x) = -\log(5-x) + 9\)[/tex].
2. For [tex]\(x\)[/tex] approaching 5 from the left ([tex]\(x \to 5^-\)[/tex]), the argument [tex]\(5 - x \to 0^+\)[/tex]. The logarithm of a value approaching zero from the positive side goes to negative infinity ([tex]\(\log(5-x) \to -\infty\)[/tex]).
3. Hence,
[tex]\[ -\log(5-x) \to +\infty \][/tex]
Then, adding 9:
[tex]\[ -\log(5-x) + 9 \to +\infty \][/tex]
This indicates that the function can attain large values as [tex]\(x\)[/tex] approaches 5 from the left.

4. For smaller values of [tex]\(x\)[/tex], as [tex]\(x\)[/tex] decreases from 5 (i.e., [tex]\( x \to -\infty \)[/tex]), [tex]\(5 - x\)[/tex] grows larger, and [tex]\(\log(5-x)\)[/tex] increases. For sufficiently large [tex]\(5 - x\)[/tex], [tex]\(\log(5-x)\)[/tex] is positive, and so [tex]\(-\log(5-x)\)[/tex] becomes negative. This implies [tex]\( -\log(5-x) + 9 < 9\)[/tex].

5. Specifically, when [tex]\(x\)[/tex] approaches [tex]\(-\infty\)[/tex], [tex]\(\log(5 - x)\)[/tex] approaches [tex]\(\log(+\infty) = +\infty\)[/tex], thus:
[tex]\[ -\log(5-x) \to -\infty \][/tex]
6. Hence,
[tex]\[ -\log(5-x) + 9 \to 9 \][/tex]
This means the function value approaches 9 from below as [tex]\(x\)[/tex] decreases without end.

Thus, the smallest value [tex]\(f(x)\)[/tex] can approach is 9, and there is no upper bound on [tex]\(f(x)\)[/tex]. Therefore, the range of [tex]\(f(x)\)[/tex] is [tex]\( y \geq 9 \)[/tex].

In conclusion:

- The domain is [tex]\( x < 5 \)[/tex].
- The range is [tex]\( y \geq 9 \)[/tex].

So, the correct answers are:

Domain: [tex]\( x < 5 \)[/tex]
Range: [tex]\( y \geq 9 \)[/tex]
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