Questions: Explain how the graph of the function f(x)=1/(x-7)^2 can be obtained from the graph of y=1/x^2. Then graph f and give the domain and range.

Transcript text: Explain how the graph of the function $f(x)=\frac{1}{(x-7)^{2}}$ can be obtained from the graph of $y=\frac{1}{x^{2}}$. Then graph $f$ and give the domain and range.

Solution

Solution Steps

Step 1: Identify the Transformation

The function $ f(x) = \frac{1}{(x-7)^2} $ can be obtained from the graph of $ y = \frac{1}{x^2} $ by shifting it 7 units to the right.

Step 2: Determine the Domain

The domain of $ f(x) = \frac{1}{(x-7)^2} $ is all real numbers except $ x = 7 $, since the function is undefined at $ x = 7 $. Therefore, the domain is $ (-\infty, 7) \cup (7, \infty) $.

Step 3: Determine the Range

The range of $ f(x) = \frac{1}{(x-7)^2} $ is $ (0, \infty) $ because the function outputs positive values for all $ x $ in its domain.

Final Answer

The graph of $ f(x) = \frac{1}{(x-7)^2} $ is obtained by shifting the graph of $ y = \frac{1}{x^2} $ 7 units to the right.
Domain: $ (-\infty, 7) \cup (7, \infty) $
Range: $ (0, \infty) $

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