Question

Graph each function. Give the domain and range. ƒ(x) = (1/3)x+2 - 1

Accepted Answer

Identify the base function and transformations. The base function here is an exponential function of the form $$f(x) = a$$^{x}$$, specifically f(x$$) = \left(\frac{1}{3}\right)^x$$. The given function is f(x$$) = -\left(\frac{1}{3}\$$right)^{x+2} - 1$, which involves transformations of the base function. Analyze the horizontal shift. The expression x + 2$ inside the exponent indicates a horizontal shift to the left by 2 units. This means the graph of $$\left(\frac{1}{3}\right)^x$$ is shifted left 2 units. Consider the reflection and vertical shift. The negative sign in front of the exponential, $$-\left(\frac{1}{3}\$$right)^{x+2}$$, reflects the graph across the x-axis. Then, subtracting 1, as in $$-\left(\frac{1}{3}\$$right)^{x+2}$$ - 1$$, shifts the graph down by 1 unit. Determine the domain. Since exponential functions are defined for all real numbers, the domain of $$f(x) is $$all real numbers, expressed as $$(-\infty, \infty). $$Find the range by considering the transformations. The base function $$\left(\frac{1}{3}\right)^x$$ has a range of $$(0, \infty). $$After reflection and vertical shift, the range becomes $$(-\infty, -1)$$ because the graph is reflected over the x-axis and shifted down by 1.

Graph each function. Give the domain and range. ƒ(x) = (1/3)^x+2 - 1

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Exponential Functions

Transformations of Functions

Domain and Range of Exponential Functions