Question

Use transformations to graph each function. Determine the domain, range, horizontal asymptote, and y-intercept of each function. $$ f(x)=2+3^{x / 2} $$

Short answer

Domain: \(-\infty < x < \infty\), Range: \(2 < y < \infty\), Horizontal Asymptote: y=2, Y-Intercept: (0, 3).

Step-by-step solution

Identify the Base Function

The base function here is the exponential function \[ f(x) = 3^{x} \]. This is a standard exponential function with base 3.

Apply Horizontal Stretch

The function \[ f(x) = 3^{x / 2} \] applies a horizontal stretch to the base function by a factor of 2. This means that the graph stretches away from the y-axis by this factor.

Apply Vertical Shift

The function \[ f(x) = 2 + 3^{x / 2} \] represents a vertical shift of the graph of \[ 3^{x/2} \] upward by 2 units. This will shift the entire graph 2 units up along the y-axis.

Determine the Domain

The domain of the function \[ f(x) = 2 + 3^{x / 2} \] is all real numbers, \(-\infty < x < \infty\), because you can input any real number into the function.

Determine the Range

The range of the function \[ f(x) = 2 + 3^{x / 2} \] is \(2 < y < \infty \), because the exponential function always outputs positive values and then shifts up by 2 units.

Determine the Horizontal Asymptote

The horizontal asymptote is y=2. This is because, as x goes to negative infinity, \(f(x)\) approaches 2 but never actually reaches it.

Find the Y-Intercept

To find the y-intercept, set x = 0 in the function \[ f(0) = 2 + 3^{0 / 2} = 2 + 1 = 3 \]. So the y-intercept is at the point (0, 3).

Key concepts

exponential functions

An exponential function is a type of mathematical function where the variable appears in the exponent. The general form is

\(f(x) = a \times b^{x}\),
where 'a' is a constant, 'b' is the base, and 'x' is the exponent. Exponential functions are known for their rapid growth or decay, depending on the base. For example, the base function in this exercise is
\(f(x) = 3^x\).
This function grows exponentially as x increases.

Transformations can be applied to these functions to shift or stretch the graph in different ways. For
example, a transformation like
\(f(x) = 3^{x/2}\) stretches the graph horizontally by a factor of 2. Adding a constant, like in
\(f(x) = 2 + 3^{x/2}\),
shifts the graph vertically.

domain and range

The domain and range of a function describe the set of possible input values (domain) and output values (range).
For exponential functions, the domain is usually all real numbers; this means


\(-finity < x < finity\).
For the function

\(f(x) = 2 + 3^{x/2}\), you can input any real number for 'x'.
The range of this exponential function, however, is limited. Since

\(3^{x/2}\)
always gives positive values and the entire graph is shifted up by 2 units, the range is


\(2 < y < finity\).
Practically, this means the function will never produce a value less than 2.

horizontal asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never quite reaches.
For the given function
\(f(x) = 2 + 3^{x/2}\),
the horizontal asymptote is y=2.
As x approaches negative infinity (
\(x \to -finity\)),
the value of

\(3^{x/2}\)
gets closer to zero. Therefore,
\f(x)
\br> approaches 2.
However, \brf(x)< never actually becomes 2. This is why y=2 is a horizontal \br> asymptote.

y-intercept

The y-intercept is the point where the graph of a function crosses the y-axis.
To find the y-intercept, you set x=0 and solve for y. For the function

\(f(x) = 2 + 3^{x/2}\),
substituting x=0 gives

\(f(0) = 2 + 3^{0/2} = 2 + 1 = 3\).
Therefore, the y-intercept occurs at the point (0, 3).
This means that when x=0, the value of the function is 3.