Math / Algebra

QuestionA function is shown below. $f(x) = \begin{cases} -x^2 + 2x & \text{for } x \le -3 \\ 2\left(\frac{1}{3}\right)^{2x} & \text{for } -3 < x < 4 \\ \frac{2x - 5}{x - 7} & \text{for } x \ge 4 \end{cases}$ Find $-3f(0) + 5(6)$

Studdy Solution

STEP 1

What is this asking? We need to plug $0$ into this crazy function, multiply the result by $-3$ , and then add that to $5$ times $6$ . Watch out! Make sure to plug $0$ into the correct piece of the function!

STEP 2

1. Evaluate $f(0)$
2. Calculate $-3f(0)$
3. Calculate $5(6)$
4. Find the final answer

STEP 3

Alright, so we've got this wild function with three different parts.
We need to figure out which part to use for $x = 0$ .
Since $0$ is between $-3$ and $4$ , we'll use the middle part of the function: $2\left(\frac{1}{3}\right)^{2x}$ .

STEP 4

Let's plug in $x = 0$ : $f(0) = 2\left(\frac{1}{3}\right)^{2 \cdot 0}$

STEP 5

Anything to the power of zero is one, so $\left(\frac{1}{3}\right)^0 = 1$ .
This makes our calculation super easy! $f(0) = 2 \cdot 1$ $f(0) = 2$

STEP 6

We just found that $f(0) = 2$ , so let's multiply that by $-3$ : $-3f(0) = -3 \cdot 2$ $-3f(0) = -6$

STEP 7

This one's a breeze! $5(6) = 5 \cdot 6$ $5(6) = 30$

STEP 8

Time to put it all together!
We have $-3f(0) = -6$ and $5(6) = 30$ , so we just need to add those together: $-3f(0) + 5(6) = -6 + 30$ $-3f(0) + 5(6) = 24$

STEP 9

Our final answer is $24$ !

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QuestionA function is shown below. $f(x) = \begin{cases} -x^2 + 2x & \text{for } x \le -3 \\ 2\left(\frac{1}{3}\right)^{2x} & \text{for } -3 < x < 4 \\ \frac{2x - 5}{x - 7} & \text{for } x \ge 4 \end{cases}$ Find $-3f(0) + 5(6)$

Studdy Solution

STEP 1

What is this asking? We need to plug $0$ into this crazy function, multiply the result by $-3$ , and then add that to $5$ times $6$ . Watch out! Make sure to plug $0$ into the correct piece of the function!

STEP 2

1. Evaluate $f(0)$
2. Calculate $-3f(0)$
3. Calculate $5(6)$
4. Find the final answer

STEP 3

Alright, so we've got this wild function with three different parts.
We need to figure out which part to use for $x = 0$ .
Since $0$ is between $-3$ and $4$ , we'll use the middle part of the function: $2\left(\frac{1}{3}\right)^{2x}$ .

STEP 4

Let's plug in $x = 0$ : $f(0) = 2\left(\frac{1}{3}\right)^{2 \cdot 0}$

STEP 5

Anything to the power of zero is one, so $\left(\frac{1}{3}\right)^0 = 1$ .
This makes our calculation super easy! $f(0) = 2 \cdot 1$ $f(0) = 2$

STEP 6

We just found that $f(0) = 2$ , so let's multiply that by $-3$ : $-3f(0) = -3 \cdot 2$ $-3f(0) = -6$

STEP 7

This one's a breeze! $5(6) = 5 \cdot 6$ $5(6) = 30$

STEP 8

Time to put it all together!
We have $-3f(0) = -6$ and $5(6) = 30$ , so we just need to add those together: $-3f(0) + 5(6) = -6 + 30$ $-3f(0) + 5(6) = 24$

STEP 9

Our final answer is $24$ !

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Studdy Solution

STEP 1

What is this asking? We need to plug 000 into this crazy function, multiply the result by −3-3−3, and then add that to 555 times 666. Watch out! Make sure to plug 000 into the *correct* piece of the function!

STEP 2

1. Evaluate f(0)f(0)f(0)2. Calculate −3f(0)-3f(0)−3f(0)3. Calculate 5(6)5(6)5(6)4. Find the final answer

STEP 3

Alright, so we've got this wild function with three different parts.We need to figure out which part to use for x=0x = 0x=0.Since 000 is between −3-3−3 and 444, we'll use the middle part of the function: 2(13)2x2\left(\frac{1}{3}\right)^{2x}2(31​)2x.

STEP 4

Let's **plug in** x=0x = 0x=0: f(0)=2(13)2⋅0f(0) = 2\left(\frac{1}{3}\right)^{2 \cdot 0}f(0)=2(31​)2⋅0

STEP 5

Anything to the power of **zero** is **one**, so (13)0=1\left(\frac{1}{3}\right)^0 = 1(31​)0=1.This makes our calculation super easy! f(0)=2⋅1f(0) = 2 \cdot 1f(0)=2⋅1 f(0)=2f(0) = 2f(0)=2

STEP 6

We just found that f(0)=2f(0) = 2f(0)=2, so let's **multiply** that by −3-3−3: −3f(0)=−3⋅2-3f(0) = -3 \cdot 2−3f(0)=−3⋅2 −3f(0)=−6-3f(0) = -6−3f(0)=−6

STEP 7

This one's a breeze! 5(6)=5⋅65(6) = 5 \cdot 65(6)=5⋅6 5(6)=305(6) = 305(6)=30

STEP 8

Time to put it all together!We have −3f(0)=−6-3f(0) = -6−3f(0)=−6 and 5(6)=305(6) = 305(6)=30, so we just need to **add** those together: −3f(0)+5(6)=−6+30-3f(0) + 5(6) = -6 + 30−3f(0)+5(6)=−6+30 −3f(0)+5(6)=24-3f(0) + 5(6) = 24−3f(0)+5(6)=24

STEP 9

Our **final answer** is 242424!

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to plug $0$ into this crazy function, multiply the result by $-3$ , and then add that to $5$ times $6$ . Watch out! Make sure to plug $0$ into the correct piece of the function!

1. Evaluate $f(0)$
2. Calculate $-3f(0)$
3. Calculate $5(6)$
4. Find the final answer

Alright, so we've got this wild function with three different parts.
We need to figure out which part to use for $x = 0$ .
Since $0$ is between $-3$ and $4$ , we'll use the middle part of the function: $2\left(\frac{1}{3}\right)^{2x}$ .

Let's plug in $x = 0$ : $f(0) = 2\left(\frac{1}{3}\right)^{2 \cdot 0}$

Anything to the power of zero is one, so $\left(\frac{1}{3}\right)^0 = 1$ .
This makes our calculation super easy! $f(0) = 2 \cdot 1$ $f(0) = 2$

We just found that $f(0) = 2$ , so let's multiply that by $-3$ : $-3f(0) = -3 \cdot 2$ $-3f(0) = -6$

This one's a breeze! $5(6) = 5 \cdot 6$ $5(6) = 30$

Time to put it all together!
We have $-3f(0) = -6$ and $5(6) = 30$ , so we just need to add those together: $-3f(0) + 5(6) = -6 + 30$ $-3f(0) + 5(6) = 24$

Our final answer is $24$ !