Which point is on the graph of $f(x)=2 \cdot 5^x$?
A. $(0,10)$
B. $(0,0)$
C. $(1,10)$
D. $(10,1)$
Answer (2)
Substitute the x-coordinate of each point into the function f ( x ) = 2 c d o t 5 x .
Evaluate f ( 0 ) = 2 c d o t 5 0 = 2 , f ( 1 ) = 2 c d o t 5 1 = 10 , and f ( 10 ) = 2 c d o t 5 10 = 19531250 .
Check which point's y-coordinate matches the function's output for its x-coordinate.
The point ( 1 , 10 ) satisfies the equation, so the answer is ( 1 , 10 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 c d o t 5 x and four points: A ( 0 , 10 ) , B ( 0 , 0 ) , C ( 1 , 10 ) , and D ( 10 , 1 ) . Our goal is to determine which of these points lies on the graph of the function. A point lies on the graph if its coordinates satisfy the function's equation.
Testing Point A Let's test each point:
Point A ( 0 , 10 ) : We need to check if f ( 0 ) = 10 . Substituting x = 0 into the function, we get: f ( 0 ) = 2 c d o t 5 0 = 2 c d o t 1 = 2 Since f ( 0 ) = 2 and not 10 , point A is not on the graph.
Testing Point B Point B ( 0 , 0 ) : We need to check if f ( 0 ) = 0 . We already found that f ( 0 ) = 2 . Since f ( 0 ) = 2 and not 0 , point B is not on the graph.
Testing Point C Point C ( 1 , 10 ) : We need to check if f ( 1 ) = 10 . Substituting x = 1 into the function, we get: f ( 1 ) = 2 c d o t 5 1 = 2 c d o t 5 = 10 Since f ( 1 ) = 10 , point C is on the graph.
Testing Point D Point D ( 10 , 1 ) : We need to check if f ( 10 ) = 1 . Substituting x = 10 into the function, we get: f ( 10 ) = 2 c d o t 5 10 = 2 c d o t 9765625 = 19531250 Since f ( 10 ) = 19531250 and not 1 , point D is not on the graph.
Conclusion Therefore, the only point that lies on the graph of f ( x ) = 2 c d o t 5 x is point C ( 1 , 10 ) .
Examples
Exponential functions like f ( x ) = 2 c d o t 5 x are used to model various real-world phenomena, such as population growth, compound interest, and radioactive decay. For example, if you invest money in an account that earns compound interest, the amount of money you have after a certain number of years can be modeled by an exponential function. Similarly, the decay of a radioactive substance can be modeled by an exponential function. Understanding how to work with exponential functions is therefore crucial in many fields, including finance, biology, and physics.
The point that lies on the graph of the function f ( x ) = 2 ⋅ 5 x is ( 1 , 10 ) . This is confirmed by evaluating the function at the given points.
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