Question

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

Accepted Answer

Identify the base function and transformations: The base function here is an exponential function of the form $$f(x) = a \cdot b$$^{x}$$, where b$$ = \frac{1}{3}$$. The given function is f(x$$) = -\left(\frac{1}{3}\$$right)^{x-2} + 2$. Notice the transformations applied to the base function. Analyze the horizontal shift: The expression x - 2$ inside the exponent indicates a horizontal shift to the right by 2 units. This means the graph of $$\left(\frac{1}{3}\right)^x$$ is shifted 2 units to the right. 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, adding 2 shifts the entire graph up by 2 units. Determine the domain: Since exponential functions are defined for all real numbers, the domain of $$f(x) is $$all real numbers, which can be written as $$(-\infty, \infty). $$Find the range: The base function $$\left(\frac{1}{3}\$$right)^{x-2}$$ is always positive. After reflection, it becomes negative, so the values are less than or equal to 0 before the vertical shift. Adding 2 shifts the range up, so the range is all real numbers less than 2, or $$(-\infty, 2)$$.

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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Exponential Functions

Transformations of Functions

Domain and Range of Functions