Graph each function. Give the domain and range. ƒ(x) = 2^x+2 - 4

Verified step by step guidance

1. Identify the base function and its transformations. The base function here is the exponential function \(f(x) = 2^x\). The given function is \(f(x) = 2^{x+2} - 4\), which involves a horizontal shift and a vertical shift.
2. Determine the horizontal shift by analyzing the exponent \(x + 2\). Since it is \(x + 2\), this represents a shift to the left by 2 units compared to the base function \$2^x$.
3. Determine the vertical shift by looking at the \(-4\) outside the exponential. This means the entire graph of \$2^{x+2}$ is shifted downward by 4 units.
4. Find the domain and range of the function. The domain of any exponential function \$2^x\( is all real numbers, so the domain of \)f(x) = 2^{x+2} - 4\( is also all real numbers. The range of the base function \)2^x$ is \((0, \, \infty)\), so after shifting down by 4, the range becomes \((-4, \, \infty)\).
5. To graph the function, start by plotting key points of the base function \$2^x\(, shift them left by 2 units, then shift down by 4 units. Draw a smooth curve through these points, remembering the horizontal asymptote is now at \)y = -4$.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the variable is in the exponent. In this question, the function is f(x) = 2^(x+2) - 4, which involves a base of 2 raised to the power (x+2). Understanding how exponential functions grow and shift is essential for graphing.

Domain and Range of Functions

The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (f(x)-values). For exponential functions like this one, the domain is usually all real numbers, while the range depends on vertical shifts and transformations.

Transformations of Functions

Transformations include shifts, stretches, and reflections applied to the base function. Here, (x+2) shifts the graph horizontally left by 2 units, and subtracting 4 shifts it vertically down by 4 units. Recognizing these helps in accurately sketching the graph and determining domain and range.