Question

use the function $f\left(x\
ight)=-\dfrac{15x}{6x + 7}$ to evaluate the statements. select the correct button in the table to show whether each statement is true or false. | | true | false | | --- | --- | --- | | the graph of $f\left(x\
ight)$ passes through the point $(0,0)$. | $\square$ | $\square$ | | the graph of $f\left(x\
ight)$ has an asymptote at $y=-\dfrac{5}{2}$ | $\square$ | $\square$ | | the range of $f\left(x\
ight)$ is all real numbers. | $\square$ | $\square$ | | the function $f\left(x\
ight)$ is even | $\square$ | $\square$ | | as $x\to+\infty$, $f\left(x\
ight)\to-\dfrac{7}{6}$ | $\square$ | $\square$ |

Explanation:

Step1: Check if (0,0) is on the graph

Substitute $x=0$ into $f(x)$:
$f(0) = -\frac{15(0)}{6(0)+7} = 0$

Step2: Find horizontal asymptote

For rational functions with equal degree numerator/denominator, asymptote is ratio of leading coefficients:
$y = -\frac{15}{6} = -\frac{5}{2}$

Step3: Determine the range

Since $y=-\frac{5}{2}$ is an asymptote, this value is not in the range, so range is not all real numbers.

Step4: Check if function is even

A function is even if $f(-x)=f(x)$. Calculate $f(-x)$:
$f(-x) = -\frac{15(-x)}{6(-x)+7} = \frac{15x}{-6x+7}$
This does not equal $f(x) = -\frac{15x}{6x+7}$, so it is not even.

Step5: Evaluate limit as $x\to+\infty$

Use leading coefficient ratio for horizontal asymptote:
$\lim_{x\to+\infty} f(x) = -\frac{15}{6} = -\frac{5}{2}
eq -\frac{7}{6}$

Answer:

1. The graph of $f(x)$ passes through the point $(0,0)$: True
2. The graph of $f(x)$ has an asymptote at $y=-\frac{5}{2}$: True
3. The range of $f(x)$ is all real numbers: False
4. The function $f(x)$ is even: False
5. As $x\to+\infty$, $f(x)\to-\frac{7}{6}$: False