use the transformations to identify the graph of the function. then

determine its
domain,
range,
and horizontal asymptote.

f(x)=4+3^X/3

To identify the graph of the function f(x) = 4 + (3^x)/3, we will analyze the transformations applied to the base function f(x) = 3^x.

1. Vertical Translation (upward): The function f(x) = 4 + (3^x)/3 is obtained by adding 4 to the base function f(x) = 3^x. This translates the graph vertically by 4 units upward.

2. Horizontal Scaling: The division by 3 in the term (3^x)/3 shrinks the graph horizontally. The effect is a compression of the graph by a factor of 1/3 compared to the base function.

Now, let's determine the domain, range, and horizontal asymptote of the given function:

1. Domain: The domain of the function f(x) is all real numbers since there are no restrictions or limitations on the exponent x.

2. Range: To determine the range, we need to consider the possible values that f(x) can take. As x approaches negative infinity, the value of 3^x approaches 0 (since the base is greater than 1), so (3^x)/3 approaches 0 as well. Adding 4 to this constant value, the range is from 4 to positive infinity: [4, ∞).

3. Horizontal Asymptote: As x approaches positive or negative infinity, the term (3^x)/3 approaches 0. Thus, the horizontal asymptote of the function f(x) = 4 + (3^x)/3 is y = 4.

In summary, the graph of the function f(x) = 4 + (3^x)/3 is a compressed version of the graph of the exponential function 3^x. The domain is all real numbers, the range is [4, ∞), and the horizontal asymptote is y = 4.

To identify the graph and determine its domain, range, and horizontal asymptote for the function f(x) = 4 + (3^x)/3, we need to understand the transformations involved.

The function f(x) = 4 + (3^x)/3 can be broken down into the following transformations:

1. Shift upward by 4 units: The "+4" in the equation indicates a vertical shift of 4 units upward.

2. Exponential growth: The exponentiation of 3^x indicates exponential growth in the y-direction.

3. Horizontal compression by a factor of 1/3: The division by 3 in the equation indicates a horizontal compression of the graph by a factor of 1/3.

Now let's determine the domain, range, and horizontal asymptote:

Domain:
Since the base of the exponential function is 3, the domain of the function f(x) = 4 + (3^x)/3 is all real numbers.

Range:
As the exponential function grows, the range of the function f(x) expands. Therefore, the range of the function f(x) = 4 + (3^x)/3 is all real numbers greater than or equal to 4.

Horizontal Asymptote:
When x approaches positive or negative infinity, the exponential term (3^x)/3 approaches zero. As a result, the horizontal asymptote of the function f(x) = 4 + (3^x)/3 is y = 4.

In summary:
- The graph of the function f(x) = 4 + (3^x)/3 is shifted 4 units upward, has exponential growth, and is horizontally compressed by a factor of 1/3.
- The domain of the function is all real numbers.
- The range of the function is all real numbers greater than or equal to 4.
- The horizontal asymptote of the function is y = 4.