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 h(x) = e^2x + 1

검증된 단계별 안내

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 inside the exponent for $h(x) = e^{2x} + 1$. The \$2x$ means a horizontal compression by a factor of \(\frac{1}{2}$ compared to \)f(x)$ because the input $x$ is multiplied by 2.

The $+1$ outside the exponential function shifts the entire graph vertically upward by 1 unit. This also shifts the horizontal asymptote from $y = 0$ to $y = 1$.

Write the equation of the asymptote explicitly: $y = 1$. This is the new horizontal asymptote for $h(x)$.

Determine the domain and range of $h(x)$. Since the exponential function is defined for all real numbers, the domain remains $(-\infty, \infty)$. The range shifts up by 1, so the new range is $(1, \infty)$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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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 a horizontal asymptote at y = 0. Understanding this base graph is essential for applying transformations.

Transformations of Functions

Transformations include shifts, stretches, compressions, and reflections applied to the base graph. For h(x) = e^(2x) + 1, the exponent 2x compresses the graph horizontally, and the +1 shifts it upward by 1 unit, changing the asymptote from y=0 to y=1. Recognizing these changes helps in sketching and analyzing the new graph.

Domain, Range, and Asymptotes of Exponential Functions

The domain of exponential functions is all real numbers, while the range depends on vertical shifts and reflections. The horizontal asymptote is a line the graph approaches but never touches, often y=0 for e^x. For h(x) = e^(2x) + 1, the asymptote shifts to y=1, and the range becomes (1, ∞).