Begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x)=1+ log₂ x

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Start by understanding the base function: $f(x) = \log_{2} x$. This function has a vertical asymptote at $x=0$, a domain of $(0, \infty)$, and a range of $(-\infty, \infty)$.

Next, analyze the given function $h(x) = 1 + \log_{2} x$. Notice that this is a vertical shift of the base function $f(x)$ upward by 1 unit.

Since vertical shifts do not affect the vertical asymptote, the vertical asymptote of $h(x)$ remains at $x=0$.

The domain of $h(x)$ is the same as the base function because the logarithm is only defined for positive $x$, so the domain is $(0, \infty)$.

The range of $h(x)$ is the range of $f(x)$ shifted up by 1, so the range is $(-\infty + 1, \infty + 1)$, which simplifies to $(-\infty, \infty)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Logarithmic Functions

Graphing logarithmic functions involves plotting points based on the logarithm's definition and understanding its shape. For f(x) = log₂ x, the graph passes through (1,0) since log base 2 of 1 is 0, and it increases slowly to the right. Recognizing this base graph is essential before applying transformations.

Transformations of Functions

Transformations modify the graph of a base function by shifting, stretching, or reflecting it. For h(x) = 1 + log₂ x, the '+1' shifts the entire graph of log₂ x upward by 1 unit. Understanding vertical shifts helps in accurately sketching the new graph and identifying changes in key features like intercepts.

Vertical Asymptotes and Domain of Logarithmic Functions

Logarithmic functions have vertical asymptotes where the argument equals zero, here at x=0. This asymptote indicates the function approaches negative infinity near this line. The domain consists of all x-values where the argument is positive, so for log₂ x, the domain is (0, ∞). Recognizing this is crucial for graphing and determining the function's range.

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