Question

Terry was asked to determine whether � ( � ) = � 3 1 � f(x)=x 3 x 1 ​ f, left parenthesis, x, right parenthesis, equals, x, cubed, plus, start fraction, 1, divided by, x, end fraction is even, odd, or neither. here is his work: step 1: find expression for � ( − � ) f(−x)f, left parenthesis, minus, x, right parenthesis � ( − � ) = ( − � ) 3 1 ( − � ) = − � 3 − 1 � f(−x) ​ =(−x) 3 (−x) 1 ​ =−x 3 − x 1 ​ ​ step 2: check if � ( − � ) f(−x)f, left parenthesis, minus, x, right parenthesis is equal to � ( � ) f(x)f, left parenthesis, x, right parenthesis or − � ( � ) −f(x)minus, f, left parenthesis, x, right parenthesis − � 3 − 1 � −x 3 − x 1 ​ minus, x, cubed, minus, start fraction, 1, divided by, x, end fraction isn't the same as � ( � ) = � 3 1 � f(x)=x 3 x 1 ​ f, left parenthesis, x, right parenthesis, equals, x, cubed, plus, start fraction, 1, divided by, x, end fraction or − � ( � ) = − � 3 1 � −f(x)=−x 3 x 1 ​ minus, f, left parenthesis, x, right parenthesis, equals, minus, x, cubed, plus, start fraction, 1, divided by, x, end fraction. step 3: conclusion � ( − � ) f(−x)f, left parenthesis, minus, x, right parenthesis isn't equivalent to either � ( � ) f(x)f, left parenthesis, x, right parenthesis or − � ( � ) −f(x)minus, f, left parenthesis, x, right parenthesis, so � ff is neither even nor odd. is terry's work correct? if not, what is the first step where terry made a mistake?

Answer 1

Terry's work is incorrect. The first step where Terry made a mistake is in Step 2 when checking if f(-x) is equal to f(x) or -f(x). The correct comparison should be f(-x) = -f(x), not f(-x) = f(x).

Step-by-step explanation:

In Step 2, Terry incorrectly compares f(-x) to both f(x) and -f(x) to determine if the function is even or odd. The correct approach is to check if f(-x) = -f(x). Let's analyze this:

Starting with the expression Terry derived for f(-x):

`\[f(-x) = -x^3 - (1)/(x)\]`

Now, compare this to -f(x):

`\[-f(x) = -(x^3 + (1)/(x)) = -x^3 - (1)/(x)\]`

The two expressions are equal, confirming that the function is odd. Terry's mistake occurred by incorrectly comparing f(-x) to f(x) and -f(x).

Therefore, the corrected conclusion is that f is an odd function. An odd function satisfies the property f(-x) = -f(x), which Terry overlooked in his analysis. It's crucial to pay attention to the proper criteria for determining whether a function is even, odd, or neither, and in this case, Terry's initial comparison was flawed.