Math / Algebra

QuestionExpress your answer as a polynomial in standard form. $\begin{array}{l} f(x)=2 x^{2}-x+6 \\ g(x)=2 x+12 \end{array}$
Find: $g(f(x))$

Studdy Solution

STEP 1

What is this asking? We're taking the function $f(x)$ and plugging it into $g(x)$ !
Function inception! Watch out! Don't forget to distribute correctly when substituting $f(x)$ into $g(x)$ !

STEP 2

1. Define the functions
2. Substitute and simplify

STEP 3

Alright, let's define our functions!
We have $f(x) = 2x^2 - x + 6$ and $g(x) = 2x + 12$ .
These are our starting points, our foundations!

STEP 4

Now, the fun part!
We need to find $g(f(x))$ , which means we're putting $f(x)$ inside $g(x)$ .
Think of it like a function turducken!

STEP 5

Wherever we see an $x$ in $g(x)$ , we replace it with the entire $f(x)$ function.
So, $g(f(x)) = 2(f(x)) + 12$ .
Let's do it!

STEP 6

We substitute the expression for $f(x)$ into $g(x)$ : $g(f(x)) = 2(2x^2 - x + 6) + 12$

STEP 7

Now, let's distribute that $\mathbf{2}$ across the terms in the parentheses: $g(f(x)) = 2 \cdot 2x^2 - 2 \cdot x + 2 \cdot 6 + 12$ $g(f(x)) = 4x^2 - 2x + 12 + 12$

STEP 8

Finally, we combine those like terms: $g(f(x)) = 4x^2 - 2x + 24$ Look at that beautiful polynomial in standard form!

STEP 9

Our final answer is $g(f(x)) = 4x^2 - 2x + 24$ .

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QuestionExpress your answer as a polynomial in standard form. $\begin{array}{l} f(x)=2 x^{2}-x+6 \\ g(x)=2 x+12 \end{array}$
Find: $g(f(x))$

Studdy Solution

STEP 1

What is this asking? We're taking the function $f(x)$ and plugging it into $g(x)$ !
Function inception! Watch out! Don't forget to distribute correctly when substituting $f(x)$ into $g(x)$ !

STEP 2

1. Define the functions
2. Substitute and simplify

STEP 3

Alright, let's define our functions!
We have $f(x) = 2x^2 - x + 6$ and $g(x) = 2x + 12$ .
These are our starting points, our foundations!

STEP 4

Now, the fun part!
We need to find $g(f(x))$ , which means we're putting $f(x)$ inside $g(x)$ .
Think of it like a function turducken!

STEP 5

Wherever we see an $x$ in $g(x)$ , we replace it with the entire $f(x)$ function.
So, $g(f(x)) = 2(f(x)) + 12$ .
Let's do it!

STEP 6

We substitute the expression for $f(x)$ into $g(x)$ : $g(f(x)) = 2(2x^2 - x + 6) + 12$

STEP 7

Now, let's distribute that $\mathbf{2}$ across the terms in the parentheses: $g(f(x)) = 2 \cdot 2x^2 - 2 \cdot x + 2 \cdot 6 + 12$ $g(f(x)) = 4x^2 - 2x + 12 + 12$

STEP 8

Finally, we combine those like terms: $g(f(x)) = 4x^2 - 2x + 24$ Look at that beautiful polynomial in standard form!

STEP 9

Our final answer is $g(f(x)) = 4x^2 - 2x + 24$ .

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QuestionExpress your answer as a polynomial in standard form. f(x)=2x2−x+6g(x)=2x+12\begin{array}{l} f(x)=2 x^{2}-x+6 \\ g(x)=2 x+12 \end{array}f(x)=2x2−x+6g(x)=2x+12​Find: g(f(x))g(f(x))g(f(x))

Studdy Solution

STEP 1

What is this asking? We're taking the function f(x)f(x)f(x) and plugging it into g(x)g(x)g(x)!Function inception! Watch out! Don't forget to distribute correctly when substituting f(x)f(x)f(x) into g(x)g(x)g(x)!

STEP 2

1. Define the functions2. Substitute and simplify

STEP 3

Alright, let's **define** our functions!We have f(x)=2x2−x+6f(x) = 2x^2 - x + 6f(x)=2x2−x+6 and g(x)=2x+12g(x) = 2x + 12g(x)=2x+12.These are our starting points, our **foundations**!

STEP 4

Now, the fun part!We need to find g(f(x))g(f(x))g(f(x)), which means we're putting f(x)f(x)f(x) *inside* g(x)g(x)g(x).Think of it like a *function turducken*!

STEP 5

Wherever we see an xxx in g(x)g(x)g(x), we **replace** it with the *entire* f(x)f(x)f(x) function.So, g(f(x))=2(f(x))+12g(f(x)) = 2(f(x)) + 12g(f(x))=2(f(x))+12.Let's do it!

STEP 6

We **substitute** the expression for f(x)f(x)f(x) into g(x)g(x)g(x): g(f(x))=2(2x2−x+6)+12g(f(x)) = 2(2x^2 - x + 6) + 12g(f(x))=2(2x2−x+6)+12

STEP 7

Now, let's **distribute** that 2\mathbf{2}2 across the terms in the parentheses: g(f(x))=2⋅2x2−2⋅x+2⋅6+12g(f(x)) = 2 \cdot 2x^2 - 2 \cdot x + 2 \cdot 6 + 12g(f(x))=2⋅2x2−2⋅x+2⋅6+12 g(f(x))=4x2−2x+12+12g(f(x)) = 4x^2 - 2x + 12 + 12g(f(x))=4x2−2x+12+12

STEP 8

Finally, we **combine** those **like terms**: g(f(x))=4x2−2x+24g(f(x)) = 4x^2 - 2x + 24g(f(x))=4x2−2x+24 Look at that beautiful polynomial in **standard form**!

STEP 9

Our **final answer** is g(f(x))=4x2−2x+24g(f(x)) = 4x^2 - 2x + 24g(f(x))=4x2−2x+24.

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Studdy solves anything!

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

QuestionExpress your answer as a polynomial in standard form. $\begin{array}{l} f(x)=2 x^{2}-x+6 \\ g(x)=2 x+12 \end{array}$
Find: $g(f(x))$

What is this asking? We're taking the function $f(x)$ and plugging it into $g(x)$ !
Function inception! Watch out! Don't forget to distribute correctly when substituting $f(x)$ into $g(x)$ !

1. Define the functions
2. Substitute and simplify

Alright, let's define our functions!
We have $f(x) = 2x^2 - x + 6$ and $g(x) = 2x + 12$ .
These are our starting points, our foundations!

Now, the fun part!
We need to find $g(f(x))$ , which means we're putting $f(x)$ inside $g(x)$ .
Think of it like a function turducken!

Wherever we see an $x$ in $g(x)$ , we replace it with the entire $f(x)$ function.
So, $g(f(x)) = 2(f(x)) + 12$ .
Let's do it!

We substitute the expression for $f(x)$ into $g(x)$ : $g(f(x)) = 2(2x^2 - x + 6) + 12$

Now, let's distribute that $\mathbf{2}$ across the terms in the parentheses: $g(f(x)) = 2 \cdot 2x^2 - 2 \cdot x + 2 \cdot 6 + 12$ $g(f(x)) = 4x^2 - 2x + 12 + 12$

Finally, we combine those like terms: $g(f(x)) = 4x^2 - 2x + 24$ Look at that beautiful polynomial in standard form!

Our final answer is $g(f(x)) = 4x^2 - 2x + 24$ .