IDNLearn.com: Your destination for reliable and timely answers to any question. Find the information you need quickly and easily with our comprehensive and accurate Q&A platform.
Sagot :
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]
### 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]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions find clarity at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
