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Chapter 1: Problem 20

Graph each function with a graphing utility using the given window. Then state the domain and range of the function. $$g(t)=\frac{1}{1+t^{2}} ; \quad[-7,7] \times[0,1.5]$$

Short Answer

Expert verified
Answer: The domain of the function is $$(-\infty, \infty)$$ and the range of the function is $$(0, 1]$$.

Step by step solution

01

Plot the function using a graphing utility

Use a graphing utility or software such as Desmos or GeoGebra to plot the given function $$g(t)=\frac{1}{1+t^2}$$ within the specified window $$[-7,7] \times [0,1.5]$$. Ensure that the plot is clear and visible.
02

Determine the domain of the function

To determine the domain of the function $$g(t)=\frac{1}{1+t^2}$$, think about what values of t make the denominator equal to zero. It is important because in the case of a rational function, the denominator should never be equal to zero, as division by zero is not defined. In this case, the denominator is $$1+t^2$$, which will never be equal to zero because squaring a real number and adding one will always result in a positive value. Therefore, the domain of g(t) is all real numbers. So, the domain can be written as: $$(-\infty,\infty)$$ or in interval notation, it is written as: $$(-\infty,\infty)$$.
03

Determine the range of the function

To determine the range of the function $$g(t)=\frac{1}{1+t^2}$$, analyze the behavior of g(t) for different values of t. As t approaches infinity or negative infinity, the denominator becomes infinitely large, and g(t) will approach zero but will never actually reach zero. On the other hand, when t is equal to zero, g(t) is equal to 1, which is the maximum value that g(t) can attain. So, g(t) will take on values between 0 (exclusive) and 1 (inclusive). Thus, the range of g(t) can be written as: $$(0,1]$$ or in interval notation as: $$(0,1]$$.

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