The range of which function is $(2, \infty)$ ?
A. $y=2^x$
B. $y=2(5^x)$
C. $y=5^{x+2}$
D. $y=5^x+2$
Answer (1)
The range of y = 2 x is ( 0 , ∞ ) .
The range of y = 2 ( 5 x ) is ( 0 , ∞ ) .
The range of y = 5 x + 2 is ( 0 , ∞ ) .
The range of y = 5 x + 2 is ( 2 , ∞ ) .
Therefore, the function with the range ( 2 , ∞ ) is y = 5 x + 2 .
Explanation
Analyzing the functions We are given four functions and we need to determine which one has the range ( 2 , ∞ ) . Let's analyze each function.
Range of y = 2 x
y = 2 x : The range of this exponential function is ( 0 , ∞ ) because 2 x is always positive for any real number x , and it can take any positive value.
Range of y = 2 ( 5 x )
y = 2 ( 5 x ) : The range of this exponential function is also ( 0 , ∞ ) because 5 x is always positive for any real number x , and multiplying it by 2 doesn't change the range, it remains positive.
Range of y = 5 x + 2
y = 5 x + 2 : The range of this exponential function is ( 0 , ∞ ) because 5 x + 2 is always positive for any real number x . We can rewrite this as y = 5 x ⋅ 5 2 = 25 c d o t 5 x , but the range is still all positive real numbers.
Range of y = 5 x + 2
y = 5 x + 2 : The range of this exponential function is ( 2 , ∞ ) because 5 x is always positive for any real number x , so 0"> 5 x > 0 . Therefore, 2"> 5 x + 2 > 2 . The function can take any value greater than 2.
Finding the matching range Comparing the ranges of the four functions with the given range ( 2 , ∞ ) , we find that the function y = 5 x + 2 has the range ( 2 , ∞ ) .
Examples
Understanding the range of exponential functions is crucial in various real-world applications, such as modeling population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns compound interest, the amount of money you have over time can be modeled by an exponential function. The range of this function tells you the possible amounts of money you can have, which is always a positive value. Similarly, in population growth, the range of the exponential function modeling the population tells you the possible population sizes, which are also always positive.
