The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = 2e^x

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Start with the base function \(f(x) = e^{x}\), which is an exponential function with a horizontal asymptote at \(y = 0\), domain \((-\infty, \infty)\), and range \((0, \infty)\).

Identify the transformation in the given function \(g(x) = 2e^{x}\). Here, the coefficient 2 is a vertical stretch factor, which means the graph of \(f(x)\) is stretched vertically by a factor of 2.

Write the equation of the transformed function explicitly: \(g(x) = 2e^{x}\). This means for each \(x\), the output is twice the value of \(e^{x}\).

Determine the asymptote of \(g(x)\). Since multiplying by 2 does not affect the horizontal asymptote, the asymptote remains \(y = 0\).

Analyze the domain and range of \(g(x)\). The domain remains all real numbers \((-\infty, \infty)\) because the exponential function is defined everywhere. The range changes due to the vertical stretch: since \(e^{x} > 0\), multiplying by 2 keeps the range as \((0, \infty)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Functions and Their Graphs

An exponential function has the form f(x) = a^x, where the base a is positive and not equal to 1. The graph of f(x) = e^x is a smooth curve increasing rapidly, passing through (0,1), with domain all real numbers and range (0, ∞). Understanding this base graph is essential before applying transformations.

Transformations of Functions

Transformations include shifts, stretches, compressions, and reflections applied to the base graph. For g(x) = 2e^x, the factor 2 vertically stretches the graph, multiplying all y-values by 2. Recognizing how these changes affect the graph and asymptotes helps in sketching and interpreting the new function.

Asymptotes, Domain, and Range

An asymptote is a line that the graph approaches but never touches. For exponential functions like e^x, the horizontal asymptote is y = 0. Transformations can shift or stretch the graph but often leave the domain as all real numbers. Identifying asymptotes and domain/range is crucial for accurate graphing.