To determine which point is on the graph of the function [tex]\( f(x) = 2 \cdot 5^x \)[/tex], we need to evaluate [tex]\(f(x)\)[/tex] at the given x-values and check which corresponding y-value matches the provided points.
Let's analyze each option step-by-step:
1. Option A: (10, 1)
- Evaluate [tex]\(f(10)\)[/tex]:
[tex]\[
f(10) = 2 \cdot 5^{10}
\][/tex]
- Calculate [tex]\(5^{10}\)[/tex]:
[tex]\[
5^{10} = 9765625
\][/tex]
- Thus:
[tex]\[
f(10) = 2 \cdot 9765625 = 19531250
\][/tex]
- Compare with y-coordinate 1:
[tex]\[
19531250 \neq 1
\][/tex]
- This point is not on the graph.
2. Option B: (0, 0)
- Evaluate [tex]\(f(0)\)[/tex]:
[tex]\[
f(0) = 2 \cdot 5^0 = 2 \cdot 1 = 2
\][/tex]
- Compare with y-coordinate 0:
[tex]\[
2 \neq 0
\][/tex]
- This point is not on the graph.
3. Option C: (1, 10)
- Evaluate [tex]\(f(1)\)[/tex]:
[tex]\[
f(1) = 2 \cdot 5^1 = 2 \cdot 5 = 10
\][/tex]
- Compare with y-coordinate 10:
[tex]\[
10 = 10
\][/tex]
- This point is on the graph.
4. Option D: (0, 10)
- Evaluate [tex]\(f(0)\)[/tex]:
[tex]\[
f(0) = 2 \cdot 5^0 = 2 \cdot 1 = 2
\][/tex]
- Compare with y-coordinate 10:
[tex]\[
2 \neq 10
\][/tex]
- This point is not on the graph.
After evaluating all the points, we find that only Option C: (1, 10) is on the graph of the function [tex]\(f(x) = 2 \cdot 5^x\)[/tex].
Thus, the correct answer is:
C. (1, 10)
