Question

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) = ex and g(x) = 2ex/2

Accepted Answer

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}}$$. Analyze the transformation from f(x$$)$$ to g(x$$)$$. The exponent changes from x$ to $$\frac{x}{2}$$, which represents a horizontal stretch by a factor of 2, and the coefficient 2 in front of the exponential represents a vertical stretch by a factor of 2. Graph f(x$$) = $$e^{x}$$ first. This graph passes through the point $$(0,1)$$, increases exponentially, and has a horizontal asymptote at $$y=0. $$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). 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 $$(0, \infty).$$

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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Exponential Functions and Their Graphs

Transformations of Functions

Asymptotes, Domain, and Range of Exponential Functions