In Exercises 5–9, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = e^x and g(x) = 2e^x/2

Verified step by step guidance

1. Identify the base function and the transformed function. Here, the base function is \(f(x) = e^{x}\), and the transformed function is \(g(x) = 2e^{\frac{x}{2}}\).
2. Analyze the transformation from \(f(x)\) to \(g(x)\). The exponent changes from \(x\) to \(\frac{x}{2}\), which represents a horizontal stretch by a factor of 2, and the coefficient 2 in front of the exponential represents a vertical stretch by a factor of 2.
3. Graph \(f(x) = e^{x}\) first. This graph passes through the point \((0,1)\), increases exponentially, and has a horizontal asymptote at \(y=0\).
4. Apply the transformations to graph \(g(x)\). Stretch the graph of \(f(x)\) horizontally by a factor of 2 (so points like \((x, e^{x})\) become \((2x, e^{x})\)), then stretch vertically by a factor of 2 (multiply the \(y\)-values by 2).
5. Determine the asymptotes, domain, and range for both functions. Both have a horizontal asymptote at \(y=0\). The domain of both functions is all real numbers \((-\infty, \infty)\). The range of \(f(x)\) is \((0, \infty)\), and since \(g(x)\) is a vertical stretch of \(f(x)\), its range is also \((0, \infty)\).

Key Concepts

Exponential Functions and Their Graphs

Exponential functions have the form f(x) = a^x, where the variable is in the exponent. The graph of f(x) = e^x is a smooth curve increasing rapidly for positive x and approaching zero for negative x. Understanding this base graph is essential for applying transformations and analyzing behavior.

Transformations of Functions

Transformations include shifts, stretches, compressions, and reflections applied to a base graph. For g(x) = 2e^(x/2), the factor 2 vertically stretches the graph, and the exponent x/2 horizontally stretches it. Recognizing these changes helps in sketching g from f and understanding how the graph changes.

Asymptotes, Domain, and Range of Exponential Functions

Exponential functions like f(x) = e^x have a horizontal asymptote, typically y = 0, which the graph approaches but never touches. The domain is all real numbers, while the range is positive real numbers. Identifying asymptotes and determining domain and range are key to fully describing the function's behavior.