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To determine which of the given points lie on the graph of [tex]\( f(x) = \log_9(x) \)[/tex], we need to check each point [tex]\((x, y)\)[/tex] to see if the given [tex]\( y \)[/tex]-coordinate matches [tex]\( \log_9(x) \)[/tex]. Let's go through each point one by one.
1. Point [tex]\( \left(-\frac{1}{81}, 2\right) \)[/tex]:
- This point has [tex]\( x = -\frac{1}{81} \)[/tex].
- Since [tex]\( \log_9(x) \)[/tex] is only defined for [tex]\( x > 0 \)[/tex] and [tex]\(-\frac{1}{81}\)[/tex] is negative, this point cannot lie on the graph of the function.
2. Point [tex]\( (0, 1) \)[/tex]:
- This point has [tex]\( x = 0 \)[/tex].
- [tex]\( \log_9(x) \)[/tex] is not defined for [tex]\( x = 0 \)[/tex] because the logarithm of 0 is undefined.
- Therefore, this point does not lie on the graph of the function.
3. Point [tex]\( \left(\frac{1}{9}, -1\right) \)[/tex]:
- This point has [tex]\( x = \frac{1}{9} \)[/tex].
- Calculate [tex]\( \log_9\left(\frac{1}{9}\right) \)[/tex]. We know that:
[tex]\[ \log_9\left(\frac{1}{9}\right) = -1 \][/tex]
- The [tex]\( y \)[/tex]-coordinate here matches [tex]\( \log_9\left(\frac{1}{9}\right) \)[/tex], so this point lies on the graph of the function.
4. Point [tex]\( (3, 243) \)[/tex]:
- This point has [tex]\( x = 3 \)[/tex].
- Calculate [tex]\( \log_9(3) \)[/tex]. We need to find [tex]\( y \)[/tex] such that [tex]\( 9^y = 3 \)[/tex].
- Recall that [tex]\( 9 = 3^2 \)[/tex], so [tex]\( (3^2)^y = 3 \implies 3^{2y} = 3 \implies 2y = 1 \implies y = \frac{1}{2} \)[/tex].
- Thus, [tex]\( \log_9(3) = \frac{1}{2} \)[/tex], not 243. Hence, this point does not lie on the graph.
5. Point [tex]\( (9, 1) \)[/tex]:
- This point has [tex]\( x = 9 \)[/tex].
- Calculate [tex]\( \log_9(9) \)[/tex]:
[tex]\[ \log_9(9) = 1 \][/tex]
- The [tex]\( y \)[/tex]-coordinate is 1, which matches [tex]\( \log_9(9) \)[/tex], so this point lies on the graph of the function.
6. Point [tex]\( (81, 2) \)[/tex]:
- This point has [tex]\( x = 81 \)[/tex].
- Calculate [tex]\( \log_9(81) \)[/tex]:
[tex]\[ \log_9(81) = 2 \quad \text{since } 9^2 = 81 \][/tex]
- The [tex]\( y \)[/tex]-coordinate here matches [tex]\( \log_9(81) \)[/tex], so this point lies on the graph of the function.
Thus, the points that lie on the graph of [tex]\(f(x) = \log_9(x) \)[/tex] are:
[tex]\[ \left( \frac{1}{9}, -1 \right), (9, 1), (81, 2) \][/tex]
1. Point [tex]\( \left(-\frac{1}{81}, 2\right) \)[/tex]:
- This point has [tex]\( x = -\frac{1}{81} \)[/tex].
- Since [tex]\( \log_9(x) \)[/tex] is only defined for [tex]\( x > 0 \)[/tex] and [tex]\(-\frac{1}{81}\)[/tex] is negative, this point cannot lie on the graph of the function.
2. Point [tex]\( (0, 1) \)[/tex]:
- This point has [tex]\( x = 0 \)[/tex].
- [tex]\( \log_9(x) \)[/tex] is not defined for [tex]\( x = 0 \)[/tex] because the logarithm of 0 is undefined.
- Therefore, this point does not lie on the graph of the function.
3. Point [tex]\( \left(\frac{1}{9}, -1\right) \)[/tex]:
- This point has [tex]\( x = \frac{1}{9} \)[/tex].
- Calculate [tex]\( \log_9\left(\frac{1}{9}\right) \)[/tex]. We know that:
[tex]\[ \log_9\left(\frac{1}{9}\right) = -1 \][/tex]
- The [tex]\( y \)[/tex]-coordinate here matches [tex]\( \log_9\left(\frac{1}{9}\right) \)[/tex], so this point lies on the graph of the function.
4. Point [tex]\( (3, 243) \)[/tex]:
- This point has [tex]\( x = 3 \)[/tex].
- Calculate [tex]\( \log_9(3) \)[/tex]. We need to find [tex]\( y \)[/tex] such that [tex]\( 9^y = 3 \)[/tex].
- Recall that [tex]\( 9 = 3^2 \)[/tex], so [tex]\( (3^2)^y = 3 \implies 3^{2y} = 3 \implies 2y = 1 \implies y = \frac{1}{2} \)[/tex].
- Thus, [tex]\( \log_9(3) = \frac{1}{2} \)[/tex], not 243. Hence, this point does not lie on the graph.
5. Point [tex]\( (9, 1) \)[/tex]:
- This point has [tex]\( x = 9 \)[/tex].
- Calculate [tex]\( \log_9(9) \)[/tex]:
[tex]\[ \log_9(9) = 1 \][/tex]
- The [tex]\( y \)[/tex]-coordinate is 1, which matches [tex]\( \log_9(9) \)[/tex], so this point lies on the graph of the function.
6. Point [tex]\( (81, 2) \)[/tex]:
- This point has [tex]\( x = 81 \)[/tex].
- Calculate [tex]\( \log_9(81) \)[/tex]:
[tex]\[ \log_9(81) = 2 \quad \text{since } 9^2 = 81 \][/tex]
- The [tex]\( y \)[/tex]-coordinate here matches [tex]\( \log_9(81) \)[/tex], so this point lies on the graph of the function.
Thus, the points that lie on the graph of [tex]\(f(x) = \log_9(x) \)[/tex] are:
[tex]\[ \left( \frac{1}{9}, -1 \right), (9, 1), (81, 2) \][/tex]
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