Math / Algebra

QuestionEvaluate the function. Round answers to four decimal places, if necessary. $\begin{array}{l} f(x)=-\frac{4}{5}(4)^{-x}+\frac{4}{5}, \text { for } f(2) \\ f(2)=\square \end{array}$ $\square$ Question Help: Written Example Submit Question

Studdy Solution

STEP 1

What is this asking? We need to plug in $x = 2$ into the function $f(x)$ and calculate the result. Watch out! Don't forget the negative sign in front of the fraction and make sure to follow the order of operations (PEMDAS/BODMAS) correctly!

STEP 2

1. Substitute the value
2. Calculate the exponent
3. Multiply the fraction
4. Add the fractions

STEP 3

Let's substitute $x = 2$ into our function $f(x) = -\frac{4}{5}(4)^{-x} + \frac{4}{5}$ .
This gives us $f(2) = -\frac{4}{5}(4)^{-2} + \frac{4}{5}$ .
Exciting stuff!

STEP 4

Remember that a negative exponent means we take the reciprocal.
So, $(4)^{-2}$ becomes $\frac{1}{(4)^{2}}$ .

STEP 5

Now, $(4)^2$ is just $4 \cdot 4 = 16$ .
So, $(4)^{-2} = \frac{1}{16}$ .

STEP 6

Our function now looks like this: $f(2) = -\frac{4}{5} \cdot \frac{1}{16} + \frac{4}{5}$ .

STEP 7

We multiply the fractions $-\frac{4}{5}$ and $\frac{1}{16}$ by multiplying the numerators together and the denominators together.
This gives us $-\frac{4 \cdot 1}{5 \cdot 16} = -\frac{4}{80}$ .

STEP 8

We can simplify the fraction $-\frac{4}{80}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 4.
This gives us $-\frac{4 \div 4}{80 \div 4} = -\frac{1}{20}$ .

STEP 9

Our function now looks like: $f(2) = -\frac{1}{20} + \frac{4}{5}$ .

STEP 10

To add the fractions, we need a common denominator.
The least common multiple of 20 and 5 is 20.

STEP 11

We can rewrite $\frac{4}{5}$ with a denominator of 20 by multiplying both the numerator and denominator by 4: $\frac{4}{5} \cdot \frac{4}{4} = \frac{16}{20}$ .

STEP 12

Now we can add the fractions: $-\frac{1}{20} + \frac{16}{20} = \frac{-1 + 16}{20} = \frac{15}{20}$ .

STEP 13

We can simplify $\frac{15}{20}$ by dividing both the numerator and denominator by 5: $\frac{15 \div 5}{20 \div 5} = \frac{3}{4}$ .

STEP 14

Finally, we convert the fraction $\frac{3}{4}$ to a decimal by dividing 3 by 4, which gives us 0.75.

STEP 15

$f(2) = 0.75$

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QuestionEvaluate the function. Round answers to four decimal places, if necessary. $\begin{array}{l} f(x)=-\frac{4}{5}(4)^{-x}+\frac{4}{5}, \text { for } f(2) \\ f(2)=\square \end{array}$ $\square$ Question Help: Written Example Submit Question

Studdy Solution

STEP 1

What is this asking? We need to plug in $x = 2$ into the function $f(x)$ and calculate the result. Watch out! Don't forget the negative sign in front of the fraction and make sure to follow the order of operations (PEMDAS/BODMAS) correctly!

STEP 2

1. Substitute the value
2. Calculate the exponent
3. Multiply the fraction
4. Add the fractions

STEP 3

Let's substitute $x = 2$ into our function $f(x) = -\frac{4}{5}(4)^{-x} + \frac{4}{5}$ .
This gives us $f(2) = -\frac{4}{5}(4)^{-2} + \frac{4}{5}$ .
Exciting stuff!

STEP 4

Remember that a negative exponent means we take the reciprocal.
So, $(4)^{-2}$ becomes $\frac{1}{(4)^{2}}$ .

STEP 5

Now, $(4)^2$ is just $4 \cdot 4 = 16$ .
So, $(4)^{-2} = \frac{1}{16}$ .

STEP 6

Our function now looks like this: $f(2) = -\frac{4}{5} \cdot \frac{1}{16} + \frac{4}{5}$ .

STEP 7

We multiply the fractions $-\frac{4}{5}$ and $\frac{1}{16}$ by multiplying the numerators together and the denominators together.
This gives us $-\frac{4 \cdot 1}{5 \cdot 16} = -\frac{4}{80}$ .

STEP 8

We can simplify the fraction $-\frac{4}{80}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 4.
This gives us $-\frac{4 \div 4}{80 \div 4} = -\frac{1}{20}$ .

STEP 9

Our function now looks like: $f(2) = -\frac{1}{20} + \frac{4}{5}$ .

STEP 10

To add the fractions, we need a common denominator.
The least common multiple of 20 and 5 is 20.

STEP 11

We can rewrite $\frac{4}{5}$ with a denominator of 20 by multiplying both the numerator and denominator by 4: $\frac{4}{5} \cdot \frac{4}{4} = \frac{16}{20}$ .

STEP 12

Now we can add the fractions: $-\frac{1}{20} + \frac{16}{20} = \frac{-1 + 16}{20} = \frac{15}{20}$ .

STEP 13

We can simplify $\frac{15}{20}$ by dividing both the numerator and denominator by 5: $\frac{15 \div 5}{20 \div 5} = \frac{3}{4}$ .

STEP 14

Finally, we convert the fraction $\frac{3}{4}$ to a decimal by dividing 3 by 4, which gives us 0.75.

STEP 15

$f(2) = 0.75$

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Studdy Solution

STEP 1

What is this asking? We need to plug in x=2x = 2x=2 into the function f(x)f(x)f(x) and calculate the result. Watch out! Don't forget the negative sign in front of the fraction and make sure to follow the order of operations (PEMDAS/BODMAS) correctly!

STEP 2

1. Substitute the value2. Calculate the exponent3. Multiply the fraction4. Add the fractions

STEP 3

Let's **substitute** x=2x = 2x=2 into our function f(x)=−45(4)−x+45f(x) = -\frac{4}{5}(4)^{-x} + \frac{4}{5}f(x)=−54​(4)−x+54​.This gives us f(2)=−45(4)−2+45f(2) = -\frac{4}{5}(4)^{-2} + \frac{4}{5}f(2)=−54​(4)−2+54​.Exciting stuff!

STEP 4

Remember that a **negative exponent** means we take the **reciprocal**.So, (4)−2(4)^{-2}(4)−2 becomes 1(4)2\frac{1}{(4)^{2}}(4)21​.

STEP 5

Now, (4)2(4)^2(4)2 is just 4⋅4=164 \cdot 4 = 164⋅4=16.So, (4)−2=116(4)^{-2} = \frac{1}{16}(4)−2=161​.

STEP 6

Our function now looks like this: f(2)=−45⋅116+45f(2) = -\frac{4}{5} \cdot \frac{1}{16} + \frac{4}{5}f(2)=−54​⋅161​+54​.

STEP 7

We multiply the fractions −45-\frac{4}{5}−54​ and 116\frac{1}{16}161​ by multiplying the **numerators** together and the **denominators** together.This gives us −4⋅15⋅16=−480-\frac{4 \cdot 1}{5 \cdot 16} = -\frac{4}{80}−5⋅164⋅1​=−804​.

STEP 8

We can **simplify** the fraction −480-\frac{4}{80}−804​ by dividing both the numerator and the denominator by their **greatest common divisor**, which is **4**.This gives us −4÷480÷4=−120-\frac{4 \div 4}{80 \div 4} = -\frac{1}{20}−80÷44÷4​=−201​.

STEP 9

Our function now looks like: f(2)=−120+45f(2) = -\frac{1}{20} + \frac{4}{5}f(2)=−201​+54​.

STEP 10

To add the fractions, we need a **common denominator**.The least common multiple of 20 and 5 is **20**.

STEP 11

We can rewrite 45\frac{4}{5}54​ with a denominator of 20 by multiplying both the numerator and denominator by 4: 45⋅44=1620\frac{4}{5} \cdot \frac{4}{4} = \frac{16}{20}54​⋅44​=2016​.

STEP 12

Now we can add the fractions: −120+1620=−1+1620=1520-\frac{1}{20} + \frac{16}{20} = \frac{-1 + 16}{20} = \frac{15}{20}−201​+2016​=20−1+16​=2015​.

STEP 13

We can **simplify** 1520\frac{15}{20}2015​ by dividing both the numerator and denominator by 5: 15÷520÷5=34\frac{15 \div 5}{20 \div 5} = \frac{3}{4}20÷515÷5​=43​.

STEP 14

Finally, we convert the fraction 34\frac{3}{4}43​ to a decimal by dividing 3 by 4, which gives us **0.75**.

STEP 15

f(2)=0.75f(2) = 0.75f(2)=0.75

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What is this asking? We need to plug in $x = 2$ into the function $f(x)$ and calculate the result. Watch out! Don't forget the negative sign in front of the fraction and make sure to follow the order of operations (PEMDAS/BODMAS) correctly!

1. Substitute the value
2. Calculate the exponent
3. Multiply the fraction
4. Add the fractions

Let's substitute $x = 2$ into our function $f(x) = -\frac{4}{5}(4)^{-x} + \frac{4}{5}$ .
This gives us $f(2) = -\frac{4}{5}(4)^{-2} + \frac{4}{5}$ .
Exciting stuff!

Remember that a negative exponent means we take the reciprocal.
So, $(4)^{-2}$ becomes $\frac{1}{(4)^{2}}$ .

Now, $(4)^2$ is just $4 \cdot 4 = 16$ .
So, $(4)^{-2} = \frac{1}{16}$ .

Our function now looks like this: $f(2) = -\frac{4}{5} \cdot \frac{1}{16} + \frac{4}{5}$ .

We multiply the fractions $-\frac{4}{5}$ and $\frac{1}{16}$ by multiplying the numerators together and the denominators together.
This gives us $-\frac{4 \cdot 1}{5 \cdot 16} = -\frac{4}{80}$ .

We can simplify the fraction $-\frac{4}{80}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 4.
This gives us $-\frac{4 \div 4}{80 \div 4} = -\frac{1}{20}$ .

Our function now looks like: $f(2) = -\frac{1}{20} + \frac{4}{5}$ .

To add the fractions, we need a common denominator.
The least common multiple of 20 and 5 is 20.

We can rewrite $\frac{4}{5}$ with a denominator of 20 by multiplying both the numerator and denominator by 4: $\frac{4}{5} \cdot \frac{4}{4} = \frac{16}{20}$ .

Now we can add the fractions: $-\frac{1}{20} + \frac{16}{20} = \frac{-1 + 16}{20} = \frac{15}{20}$ .

We can simplify $\frac{15}{20}$ by dividing both the numerator and denominator by 5: $\frac{15 \div 5}{20 \div 5} = \frac{3}{4}$ .

Finally, we convert the fraction $\frac{3}{4}$ to a decimal by dividing 3 by 4, which gives us 0.75.

$f(2) = 0.75$