QuestionFind the domain of the function $P(t)=\frac{\sqrt{t-5}}{2 t-14}$ . What is the domain?

Studdy Solution

STEP 1

Assumptions1. The function is $(t)=\frac{\sqrt{t-5}}{ t-14}$ . We are looking for the domain of this function, which is the set of all possible values of $t$ that make the function defined.
3. The function is undefined when the denominator is zero or when the radicand (the number under the square root) is negative.

STEP 2

First, let's find the values of $t$ that make the denominator zero. We set the denominator equal to zero and solve for $t$ .
$2 t -14 =0$

STEP 3

olve the equation for $t$ .
$t = \frac{14}{2}$

STEP 4

Calculate the value of $t$ .
$t =7$

STEP 5

So, the function is undefined when $t =7$ . Now, let's find the values of $t$ that make the radicand negative. We set the radicand greater than or equal to zero and solve for $t$ .
$t -5 \geq0$

STEP 6

olve the inequality for $t$ .
$t \geq5$

STEP 7

So, the function is defined when $t \geq5$ . However, we must exclude $t =7$ from the domain because it makes the denominator zero.

STEP 8

Therefore, the domain of the function is $t \geq5$ and $t \neq7$ . In interval notation, this is $[5,7) \cup (7, \infty)$ .
The domain of the function is $[5,7) \cup (7, \infty)$ .

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QuestionFind the domain of the function $P(t)=\frac{\sqrt{t-5}}{2 t-14}$ . What is the domain?

Studdy Solution

STEP 1

STEP 2

First, let's find the values of $t$ that make the denominator zero. We set the denominator equal to zero and solve for $t$ .
$2 t -14 =0$

STEP 3

olve the equation for $t$ .
$t = \frac{14}{2}$

STEP 4

Calculate the value of $t$ .
$t =7$

STEP 5

So, the function is undefined when $t =7$ . Now, let's find the values of $t$ that make the radicand negative. We set the radicand greater than or equal to zero and solve for $t$ .
$t -5 \geq0$

STEP 6

olve the inequality for $t$ .
$t \geq5$

STEP 7

So, the function is defined when $t \geq5$ . However, we must exclude $t =7$ from the domain because it makes the denominator zero.

STEP 8

Therefore, the domain of the function is $t \geq5$ and $t \neq7$ . In interval notation, this is $[5,7) \cup (7, \infty)$ .
The domain of the function is $[5,7) \cup (7, \infty)$ .

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QuestionFind the domain of the function P(t)=t−52t−14P(t)=\frac{\sqrt{t-5}}{2 t-14}P(t)=2t−14t−5​​. What is the domain?

Studdy Solution

STEP 1

STEP 2

First, let's find the values of ttt that make the denominator zero. We set the denominator equal to zero and solve for ttt.2t−14=02 t -14 =02t−14=0

STEP 3

olve the equation for ttt.t=142t = \frac{14}{2}t=214​

STEP 4

Calculate the value of ttt.t=7t =7t=7

STEP 5

So, the function is undefined when t=7t =7t=7. Now, let's find the values of ttt that make the radicand negative. We set the radicand greater than or equal to zero and solve for ttt.t−5≥0t -5 \geq0t−5≥0

STEP 6

olve the inequality for ttt.t≥5t \geq5t≥5

STEP 7

So, the function is defined when t≥5t \geq5t≥5. However, we must exclude t=7t =7t=7 from the domain because it makes the denominator zero.

STEP 8

Therefore, the domain of the function is t≥5t \geq5t≥5 and t≠7t \neq7t=7. In interval notation, this is [5,7)∪(7,∞)[5,7) \cup (7, \infty)[5,7)∪(7,∞).The domain of the function is [5,7)∪(7,∞)[5,7) \cup (7, \infty)[5,7)∪(7,∞).

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QuestionFind the domain of the function $P(t)=\frac{\sqrt{t-5}}{2 t-14}$ . What is the domain?

First, let's find the values of $t$ that make the denominator zero. We set the denominator equal to zero and solve for $t$ .
$2 t -14 =0$

olve the equation for $t$ .
$t = \frac{14}{2}$

Calculate the value of $t$ .
$t =7$

So, the function is undefined when $t =7$ . Now, let's find the values of $t$ that make the radicand negative. We set the radicand greater than or equal to zero and solve for $t$ .
$t -5 \geq0$

olve the inequality for $t$ .
$t \geq5$

So, the function is defined when $t \geq5$ . However, we must exclude $t =7$ from the domain because it makes the denominator zero.

Therefore, the domain of the function is $t \geq5$ and $t \neq7$ . In interval notation, this is $[5,7) \cup (7, \infty)$ .
The domain of the function is $[5,7) \cup (7, \infty)$ .