What is the range of [tex]f(x)=3^x+9[/tex]?
A. [tex]{y \mid y<9}[/tex]
B. [tex]{y \mid y>9}[/tex]
C. [tex]{y \mid y>3}[/tex]
D. [tex]{y \mid y<3}[/tex]
Answer (2)
The function is f ( x ) = 3 x + 9 .
The exponential term 3 x is always positive, so 0"> 3 x > 0 .
Therefore, 9"> f ( x ) = 3 x + 9 > 9 .
The range of f ( x ) is 9}"> y ∣ y > 9 . 9}}"> y ∣ y > 9
Explanation
Understanding the Problem We are asked to find the range of the function f ( x ) = 3 x + 9 . The range of a function is the set of all possible output values (y-values) that we can get from the function.
Analyzing the Function Let's analyze the function f ( x ) = 3 x + 9 . It consists of two parts: an exponential term 3 x and a constant term 9 .
Analyzing the Exponential Term The exponential function 3 x is always positive for any real number x . As x approaches negative infinity, 3 x approaches 0, but it never actually reaches 0. As x approaches positive infinity, 3 x also approaches infinity. Therefore, 0"> 3 x > 0 for all x .
Determining the Lower Bound Now, let's consider the entire function f ( x ) = 3 x + 9 . Since 0"> 3 x > 0 , we can say that 0 + 9"> f ( x ) > 0 + 9 , which means 9"> f ( x ) > 9 . So, the function's value is always greater than 9.
Determining the Upper Bound As x approaches infinity, 3 x approaches infinity, so f ( x ) = 3 x + 9 also approaches infinity. This means there is no upper bound for the function.
Expressing the Range Therefore, the range of the function f ( x ) = 3 x + 9 is all real numbers greater than 9. In set notation, this is written as 9}"> y ∣ y > 9 .
Examples
Understanding the range of exponential functions like f ( x ) = 3 x + 9 is crucial in various real-world scenarios. For instance, consider modeling population growth or the spread of a virus. The exponential term 3 x represents the rapid increase, while the constant term 9 might represent an initial population or number of infected individuals. Knowing the range helps us predict the possible future values and understand the limitations of the model. For example, we know the population will always be greater than 9, and it can grow indefinitely.
The range of the function f ( x ) = 3 x + 9 is all real numbers greater than 9, expressed as 9 \}"> { y ∣ y > 9 } . Therefore, the correct answer is option B. The function is defined for all x , and it grows indefinitely as x increases.
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