What are the domain and range of [tex]f(x)=\left(\frac{1}{6}\right)^x+2[/tex]?
A. domain: [tex]\left\{x \left\lvert\, x>-\frac{1}{6}\right.\right\}[/tex]; range: [tex]\{y \mid y>0\}[/tex]
B. domain: [tex]\left\{x \left\lvert\, x>\frac{1}{6}\right.\right\}[/tex]; range: [tex]\{y \mid y>2\}[/tex]
C. domain: [tex]\{x \mid x[/tex] is a real number [tex]\}[/tex]; range: [tex]\{y \mid y>2\}[/tex]
D. domain: [tex]\{x \mid x[/tex] is a real number [tex]\}[/tex]; range: [tex]\{y \mid y>-2\}[/tex]
Answer (2)
The domain of the exponential function ( 6 1 ) x is all real numbers.
Adding 2 to the function does not change the domain, so the domain of f ( x ) is all real numbers.
The range of the exponential function ( 6 1 ) x is ( 0 , ∞ ) .
Adding 2 to the function shifts the range up by 2, so the range of f ( x ) is ( 2 , ∞ ) .
Therefore, the domain is { x ∣ x is a real number } and the range is 2\}"> { y ∣ y > 2 } .
The final answer is 2\}}"> domain: { x ∣ x is a real number } ; range: { y ∣ y > 2 }
Explanation
Analyzing the Function We are asked to find the domain and range of the function f ( x ) = ( 6 1 ) x + 2 . Let's analyze the function to determine these properties.
Determining the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. For an exponential function of the form a x , where 0"> a > 0 , the domain is all real numbers. Since 0"> 6 1 > 0 , the domain of ( 6 1 ) x is all real numbers. Adding a constant, 2 in this case, does not change the domain. Therefore, the domain of f ( x ) = ( 6 1 ) x + 2 is all real numbers.
Determining the Range The range of a function is the set of all possible output values (y-values) that the function can produce. For an exponential function of the form a x , where 0"> a > 0 , the range is ( 0 , ∞ ) . This means that ( 6 1 ) x can take any positive value. Adding 2 to the function shifts the entire range up by 2 units. Therefore, the range of f ( x ) = ( 6 1 ) x + 2 is ( 2 , ∞ ) , which means 2"> y > 2 .
Final Answer In set notation, the domain is written as { x ∣ x is a real number } and the range is written as 2\right\}"> { y ∣ y > 2 } . Therefore, the correct answer is: domain: { x ∣ x is a real number } ; range: 2\}"> { y ∣ y > 2 }
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account that pays compound interest, the amount of money you have in the account after a certain period of time can be modeled by an exponential function. Understanding the domain and range of these functions can help you make informed decisions about your investments.
The domain of the function f ( x ) = ( 6 1 ) x + 2 is all real numbers, represented as { x ∣ x is a real number } . The range is all y-values greater than 2, represented as 2\}"> { y ∣ y > 2 } . Therefore, the correct answer is option C.
;
