Graph each function. Give the domain and range. $\text{ƒ}\left(x\right)=2^{(x-1)}+2$

Verified step by step guidance

1. Identify the given function: \(ƒ(x) = 2^{\left(x - 1\right)} + 2\). This is an exponential function with base 2, shifted horizontally and vertically.
2. Determine the horizontal shift by examining the exponent \(x - 1\). The graph of \$2^x\( is shifted 1 unit to the right because of the \)-1$ inside the exponent.
3. Determine the vertical shift by looking at the \(+2\) outside the exponential expression. This shifts the entire graph 2 units upward.
4. Find the domain of the function. Since exponential functions are defined for all real numbers, the domain is all real numbers, expressed as \((-\infty, \infty)\).
5. Find the range of the function. The base function \$2^x\( has a range of \)(0, \infty)\(, but due to the vertical shift of +2, the range becomes \)(2, \infty)$.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the variable is in the exponent. These functions model rapid growth or decay and have distinctive curves that never touch the x-axis. Understanding their shape and behavior is essential for graphing and analyzing transformations.

Function Transformations

Transformations modify the graph of a base function by shifting, stretching, compressing, or reflecting it. For f(x) = 2^(x-1) + 2, the (x-1) shifts the graph right by 1 unit, and the +2 shifts it up by 2 units. Recognizing these changes helps in accurately sketching the graph.

Domain and Range of Exponential Functions

The domain of exponential functions is all real numbers since any real input is valid. The range depends on vertical shifts; for f(x) = 2^(x-1) + 2, the output is always greater than 2, so the range is (2, ∞). Identifying domain and range is key to understanding function behavior.