use the transformations to identify the graph of the function. then
determine its
domain,
range,
and horizontal asymptote.
f(x)=4+3^X/3
please show work
To identify the graph of the function f(x) = 4 + (3^x)/3 using transformations, we will break down the function into its components.
The original function is 3^x, which is an exponential function with a base of 3. This means that the graph starts at (0, 1) and grows rapidly as x increases.
1. Translation (Vertical shift):
The function 3^x is transformed by adding 4 to the entire function: f(x) = 4 + 3^x. This results in a vertical shift of 4 units upwards, so the graph is shifted upwards by 4 units.
2. Stretching (Vertical scaling):
Next, we divide the function by 3: f(x) = (3^x)/3. This scaling factor of 1/3 compresses the graph vertically. It makes the exponential growth slower than the original function.
Taking these transformations into account, we can identify the graph of the function f(x) = 4 + (3^x)/3.
Now, let's determine the domain, range, and horizontal asymptote of the function:
Domain:
The domain is the set of all possible x-values for which the function is defined. Since the exponential function 3^x is defined for all real numbers x, we can conclude that the domain of f(x) = 4 + (3^x)/3 is also all real numbers.
Range:
The range is the set of all possible y-values the function can take. Since the exponential function 3^x is always positive (as long as x is real), dividing it by 3 and adding 4 does not change the positivity. Therefore, the range of f(x) = 4 + (3^x)/3 is y ≥ 4.
Horizontal Asymptote:
To find the horizontal asymptote, we examine the behavior of the function as x approaches positive and negative infinity. In this case, the function has a horizontal asymptote at y = 0. This means that as x becomes very large or very small, the values of f(x) get closer and closer to 0.
In summary:
- The graph of the function f(x) = 4 + (3^x)/3 is obtained by shifting the graph of 3^x upwards by 4 units and compressing it vertically by a factor of 1/3.
- The domain of the function is all real numbers.
- The range of the function is y ≥ 4.
- The horizontal asymptote of the function is y = 0.