Math / Algebra

QuestionThe function $f$ is defined by $f(x)=x^{2}+3$ . Find $f(5 z)$ . $f(5 z)=$

Studdy Solution

STEP 1

What is this asking? Given a function $f(x)$ that squares its input and adds 3, what's the result when the input is $5z$ ? Watch out! Don't forget to square the entire input $5z$ , not just the $z$ !

STEP 2

1. Substitute the Input
2. Expand and Simplify

STEP 3

Alright, so we're given the function $f(x) = x^2 + 3$ .
This tells us that whatever we put inside the parentheses, we square it and then add 3.
Our goal is to find $f(5z)$ .

STEP 4

Let's substitute $5z$ for $x$ in our function.
So, wherever we see an $x$ , we're going to replace it with $5z$ .
This gives us $f(5z) = (5z)^2 + 3$ .
Notice how the entire $5z$ is being squared!

STEP 5

Now, let's expand $(5z)^2$ .
Remember, $(5z)^2$ means $(5z) \cdot (5z)$ .

STEP 6

When we multiply, we get $5 \cdot z \cdot 5 \cdot z$ .
We can rearrange this as $5 \cdot 5 \cdot z \cdot z$ .

STEP 7

$5 \cdot 5$ gives us 25, and $z \cdot z$ gives us $z^2$ .
So, $(5z)^2$ simplifies to $25z^2$ .

STEP 8

Putting it all together, we have $f(5z) = 25z^2 + 3$ .
And that's it!

STEP 9

$f(5z) = 25z^2 + 3$

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QuestionThe function $f$ is defined by $f(x)=x^{2}+3$ . Find $f(5 z)$ . $f(5 z)=$

Studdy Solution

STEP 1

What is this asking? Given a function $f(x)$ that squares its input and adds 3, what's the result when the input is $5z$ ? Watch out! Don't forget to square the entire input $5z$ , not just the $z$ !

STEP 2

1. Substitute the Input
2. Expand and Simplify

STEP 3

Alright, so we're given the function $f(x) = x^2 + 3$ .
This tells us that whatever we put inside the parentheses, we square it and then add 3.
Our goal is to find $f(5z)$ .

STEP 4

Let's substitute $5z$ for $x$ in our function.
So, wherever we see an $x$ , we're going to replace it with $5z$ .
This gives us $f(5z) = (5z)^2 + 3$ .
Notice how the entire $5z$ is being squared!

STEP 5

Now, let's expand $(5z)^2$ .
Remember, $(5z)^2$ means $(5z) \cdot (5z)$ .

STEP 6

When we multiply, we get $5 \cdot z \cdot 5 \cdot z$ .
We can rearrange this as $5 \cdot 5 \cdot z \cdot z$ .

STEP 7

$5 \cdot 5$ gives us 25, and $z \cdot z$ gives us $z^2$ .
So, $(5z)^2$ simplifies to $25z^2$ .

STEP 8

Putting it all together, we have $f(5z) = 25z^2 + 3$ .
And that's it!

STEP 9

$f(5z) = 25z^2 + 3$

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QuestionThe function fff is defined by f(x)=x2+3f(x)=x^{2}+3f(x)=x2+3. Find f(5z)f(5 z)f(5z). f(5z)=f(5 z)=f(5z)=

Studdy Solution

STEP 1

What is this asking? Given a function f(x)f(x)f(x) that squares its input and adds 3, what's the result when the input is 5z5z5z? Watch out! Don't forget to square the *entire* input 5z5z5z, not just the zzz!

STEP 2

1. Substitute the Input2. Expand and Simplify

STEP 3

Alright, so we're given the function f(x)=x2+3f(x) = x^2 + 3f(x)=x2+3.This tells us that whatever we put inside the parentheses, we **square it** and then **add 3**.Our goal is to find f(5z)f(5z)f(5z).

STEP 4

Let's **substitute** 5z5z5z for xxx in our function.So, wherever we see an xxx, we're going to replace it with 5z5z5z.This gives us f(5z)=(5z)2+3f(5z) = (5z)^2 + 3f(5z)=(5z)2+3.Notice how the *entire* 5z5z5z is being squared!

STEP 5

Now, let's **expand** (5z)2(5z)^2(5z)2.Remember, (5z)2(5z)^2(5z)2 means (5z)⋅(5z)(5z) \cdot (5z)(5z)⋅(5z).

STEP 6

When we multiply, we get 5⋅z⋅5⋅z5 \cdot z \cdot 5 \cdot z5⋅z⋅5⋅z.We can rearrange this as 5⋅5⋅z⋅z5 \cdot 5 \cdot z \cdot z5⋅5⋅z⋅z.

STEP 7

5⋅55 \cdot 55⋅5 gives us **25**, and z⋅zz \cdot zz⋅z gives us z2z^2z2.So, (5z)2(5z)^2(5z)2 simplifies to 25z225z^225z2.

STEP 8

Putting it all together, we have f(5z)=25z2+3f(5z) = 25z^2 + 3f(5z)=25z2+3.And that's it!

STEP 9

f(5z)=25z2+3f(5z) = 25z^2 + 3f(5z)=25z2+3

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QuestionThe function $f$ is defined by $f(x)=x^{2}+3$ . Find $f(5 z)$ . $f(5 z)=$

What is this asking? Given a function $f(x)$ that squares its input and adds 3, what's the result when the input is $5z$ ? Watch out! Don't forget to square the entire input $5z$ , not just the $z$ !

1. Substitute the Input
2. Expand and Simplify

Alright, so we're given the function $f(x) = x^2 + 3$ .
This tells us that whatever we put inside the parentheses, we square it and then add 3.
Our goal is to find $f(5z)$ .

Let's substitute $5z$ for $x$ in our function.
So, wherever we see an $x$ , we're going to replace it with $5z$ .
This gives us $f(5z) = (5z)^2 + 3$ .
Notice how the entire $5z$ is being squared!

Now, let's expand $(5z)^2$ .
Remember, $(5z)^2$ means $(5z) \cdot (5z)$ .

When we multiply, we get $5 \cdot z \cdot 5 \cdot z$ .
We can rearrange this as $5 \cdot 5 \cdot z \cdot z$ .

$5 \cdot 5$ gives us 25, and $z \cdot z$ gives us $z^2$ .
So, $(5z)^2$ simplifies to $25z^2$ .

Putting it all together, we have $f(5z) = 25z^2 + 3$ .
And that's it!

$f(5z) = 25z^2 + 3$