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)
Evaluate the function f ( x ) = 2 c d o t 5 x at x = 0 and find that f ( 0 ) = 2 , which means the point (0, 2) is on the graph.
Evaluate the function at x = 1 and find that f ( 1 ) = 10 , which means the point (1, 10) is on the graph.
Evaluate the function at x = 10 and find that f ( 10 ) = 19531250 , which means the point (10, 19531250) is on the graph.
Therefore, the point on the graph of f ( x ) = 2 c d o t 5 x is ( 1 , 10 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = 2 ⋅ 5 x and four points: A(0, 10), B(0, 0), C(1, 10), and D(10, 1). We need 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.
Checking Each Point Let's check each point:
Point A (0, 10): Substitute x = 0 into the function: f ( 0 ) = 2 ⋅ 5 0 = 2 ⋅ 1 = 2 . Since f ( 0 ) = 2 = 10 , point A is not on the graph.
Point B (0, 0): Substitute x = 0 into the function: f ( 0 ) = 2 ⋅ 5 0 = 2 ⋅ 1 = 2 . Since f ( 0 ) = 2 = 0 , point B is not on the graph.
Point C (1, 10): Substitute x = 1 into the function: f ( 1 ) = 2 ⋅ 5 1 = 2 ⋅ 5 = 10 . Since f ( 1 ) = 10 , point C is on the graph.
Point D (10, 1): Substitute x = 10 into the function: f ( 10 ) = 2 ⋅ 5 10 = 2 ⋅ 9765625 = 19531250 . Since f ( 10 ) = 19531250 = 1 , point D is not on the graph.
Conclusion Therefore, only point C (1, 10) satisfies the equation f ( x ) = 2 ⋅ 5 x .
Examples
Exponential functions like f ( x ) = 2 ⋅ 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 essential in many fields, including finance, biology, and physics.
The point on the graph of the function f ( x ) = 2 ⋅ 5 x is C (1, 10) because evaluating the function at x = 1 gives f ( 1 ) = 10 . Other candidate points do not satisfy the function's equation.
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