Questions: Determine whether the equation defines (y) as a function of (x). [ y=frac9x ] Does the equation define (y) as a function of (x) ? Yes No

$Determine whether the equation defines (y) as a function of (x). [ y=frac9x ] Does the equation define (y) as a function of (x) ? Yes No$

Transcript text: Determine whether the equation defines $y$ as a function of $x$. \[ y=\frac{9}{x} \] Does the equation define $y$ as a function of $x$ ? Yes No

Solution

Solution Steps

To determine if the equation defines $ y $ as a function of $ x $, we need to check if for every value of $ x $, there is exactly one corresponding value of $ y $. In this case, the equation $ y = \frac{9}{x} $ is a rational function, which means for every non-zero $ x $, there is exactly one $ y $. However, $ x $ cannot be zero because division by zero is undefined. Therefore, $ y $ is a function of $ x $ for all $ x \neq 0 $.

Step 1: Determine Functionality

To determine if the equation $ y = \frac{9}{x} $ defines $ y $ as a function of $ x $, we need to check if each $ x $ value corresponds to exactly one $ y $ value.

Step 2: Analyze Values

We tested several values of $ x $:

For $ x = -10 $: $ y = \frac{9}{-10} = -0.9 $ (True)
For $ x = -1 $: $ y = \frac{9}{-1} = -9 $ (True)
For $ x = 0 $: Division by zero is undefined (False)
For $ x = 1 $: $ y = \frac{9}{1} = 9 $ (True)
For $ x = 10 $: $ y = \frac{9}{10} = 0.9 $ (True)

Step 3: Conclusion

Since $ y $ is defined for all $ x \neq 0 $ and each $ x $ corresponds to exactly one $ y $, we conclude that the equation does define $ y $ as a function of $ x $.