Question

Graph each function. Give the domain and range. $$f(x)=2^{x+2}-4$$

   Graph each function. Give the domain and range.
$$f(x)=2^{x+2}-4$$

Precalculus

Margaret L. Lial,… 6th Edition

Chapter 4, Problem 49

Instant Answer

Solved by Expert Nick Johnson

05/22/2021

Step 1

Step 1: First, we need to understand the function $$f(x)=2^{x+2}-4$$ This function is a transformation of the base function $$f(x)=2^x$$ The "+2" inside the exponent shifts the graph 2 units to the left, and the "-4" outside the exponent shifts the graph 4 units Show more…

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Graph each function. Give the domain and range. $$f(x)=2^{x+2}-4$$

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Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. These functions exhibit rapid growth or decay and have the general form f(x) = a^(x) where a > 0 and a ? 1. They are foundational in modeling real-world phenomena such as population growth, radioactive decay, and financial investments.

Function Transformations

Function transformations involve shifting, stretching, compressing, or reflecting the graph of a basic function to obtain a new graph. In the case of exponential functions, a horizontal shift (such as replacing x with x+c) moves the graph left or right, and a vertical shift (such as subtracting a constant) moves the graph up or down. These transformations allow for the adaptation of the basic exponential curve to fit various situations.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For exponential functions, the domain is typically all real numbers. However, the range can change depending on vertical shifts or reflections. Understanding domain and range is essential to fully describing the behavior and limitations of any function.

Graphing of Functions

Graphing functions involves plotting key features such as intercepts, asymptotes, and the effects of any transformations. With exponential functions, the graph typically shows a smooth, continuous curve that never touches its horizontal asymptote. Recognizing these graph characteristics is essential for visually analyzing the function's behavior.

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