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Chapter 4: Problem 91

Use a graphing utility to graph the function. $$f(x)=\arctan (2 x-3)$$

Short Answer

Expert verified
The graph of the function \(f(x) = \arctan(2x - 3)\) would look like the arctan function, but compressed horizontally by a factor of 1/2 and shifted to the right by 3 units.

Step by step solution

01

Understand the base function

The base function in this exercise is the arctan function. The arctan function, also known as the inverse tangent function, gives the angle whose tangent is the input value. The graph of the base arctan function would start near -蟺/2 for very negative x, cross through (0,0), and approach 蟺/2 as x gets very large.
02

Understand the transformation

Here, instead of feeding x directly into the function as input, we're feeding `2x - 3` into the function. This represents changing the input to the base function by first multiplying by 2 (which would horizontally compress the graph by a factor of 1/2) and then subtracting 3 (which would shift the graph to the right by 3 units).
03

Use graphing utility to graph the function

Most graphing utilities will let you input the function directly. Input the function \(f(x) = \arctan(2x - 3)\) and observe the transformed graph. It should still cross the y-axis at (0,0) and have the general shape of the arctan function, but it will be compressed horizontally and shifted to the right compared to the base function.

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