In Exercises 47–52, graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = (½)^x and g(x) = (1/2)^(x-1) + 1

Verified step by step guidance

1. Identify the base functions and their transformations. The function \(f(x) = (\frac{1}{2})^x\) is an exponential decay function with base \(\frac{1}{2}\), and \(g(x) = (\frac{1}{2})^{x-1} + 1\) is a horizontal shift and vertical shift of \(f(x)\).
2. Determine the asymptotes for each function. For \(f(x)\), the horizontal asymptote is \(y = 0\) because as \(x \to \infty\), \((\frac{1}{2})^x \to 0\). For \(g(x)\), the vertical shift by \(+1\) moves the horizontal asymptote to \(y = 1\).
3. Graph \(f(x)\) by plotting key points such as \(f(0) = 1\), \(f(1) = \frac{1}{2}\), and \(f(-1) = 2\), and sketch the curve approaching the asymptote \(y=0\) as \(x\) increases.
4. Graph \(g(x)\) by applying the horizontal shift of 1 unit to the right (replace \(x\) by \(x-1\)) and then shifting the entire graph up by 1 unit. Plot points like \(g(1) = 1 + (\frac{1}{2})^0 = 2\), \(g(2) = 1 + (\frac{1}{2})^1 = 1.5\), and \(g(0) = 1 + (\frac{1}{2})^{-1} = 3\) to help sketch the curve.
5. Confirm the graphs and asymptotes using a graphing utility by entering both functions and observing their behavior, ensuring the asymptotes \(y=0\) for \(f(x)\) and \(y=1\) for \(g(x)\) are correctly represented.

Key Concepts

Exponential Functions

Exponential functions have the form f(x) = a^x, where the base a is a positive constant. They exhibit rapid growth or decay depending on whether a is greater or less than 1. Understanding their shape and behavior is essential for graphing and comparing functions like f(x) = (1/2)^x and g(x) = (1/2)^{x-1} + 1.

Transformations of Functions

Function transformations include shifts, stretches, and reflections that alter the graph's position or shape. For example, g(x) = (1/2)^{x-1} + 1 is a horizontal shift right by 1 unit and a vertical shift up by 1 unit of f(x) = (1/2)^x. Recognizing these helps in sketching graphs and understanding their relationships.

Asymptotes of Functions

Asymptotes are lines that a graph approaches but never touches. Exponential functions often have horizontal asymptotes representing limits as x approaches infinity or negative infinity. Identifying and writing equations of asymptotes is crucial for accurately graphing and analyzing functions like f and g.