$y=4^{x-5}+3$
The domain of this function is
$\square$ .
The range of this function is
$\square$
Answer (1)
The domain of the function y = 4 x − 5 + 3 is all real numbers.
The exponential term 4 x − 5 is always positive.
Therefore, the range of the function is 3"> y > 3 .
The domain is all real numbers and the range is all real numbers greater than 3: 3}"> y > 3 .
Explanation
Analyzing the Function Let's analyze the given function to determine its domain and range. The function is: y = 4 x − 5 + 3 We need to find the set of all possible x values (domain) and the set of all possible y values (range).
Determining the Domain To find the domain, we need to identify any restrictions on the x values. Since the exponent x − 5 can be any real number, there are no restrictions on x . Therefore, the domain is all real numbers.
Analyzing the Range Now, let's find the range. The exponential term 4 x − 5 is always positive for any real number x . That is, 0"> 4 x − 5 > 0 for all x .
Determining the Range Since 0"> 4 x − 5 > 0 , we can say that 0 + 3"> y = 4 x − 5 + 3 > 0 + 3 , which means 3"> y > 3 . Therefore, the range of the function is all real numbers greater than 3.
Final Answer In conclusion, the domain of the function y = 4 x − 5 + 3 is all real numbers, and the range is all real numbers greater than 3.
Examples
Understanding the domain and range of functions is crucial in many real-world applications. For example, consider a scenario where the function represents the growth of a bacteria population over time. The domain would represent the time frame during which the population is observed, and the range would represent the possible sizes of the bacteria population. Knowing the domain and range helps scientists make accurate predictions and understand the limitations of their models. Another example is in physics, where the domain could represent the possible angles of a projectile launch, and the range could represent the possible distances the projectile can travel. Understanding these concepts allows for accurate calculations and predictions in various scientific fields.
