Math / Algebra

QuestionFor the given functions, $f(x) = x^2 + 3$ and $g(x) = 5x - 3$ , find the indicated composition. Write your answer by filling-in the blanks.
a. $(f \circ g)(x) =$
b. $(f \circ g)(4) =$
Moving to another question will save this response.
Question 21 of 23

Studdy Solution

STEP 1

What is this asking? We need to find the composition of two functions, $f(x)$ and $g(x)$ , both when $x$ is a variable and when $x$ is the number $4$ . Watch out! Function composition isn't just multiplying the functions!
It's about plugging one function into the other.

STEP 2

1. Find $(f \circ g)(x)$
2. Find $(f \circ g)(4)$

STEP 3

Alright, let's start with what $(f \circ g)(x)$ even means.
It means we take $g(x)$ and plug it in wherever we see an $x$ in $f(x)$ .
It's like a function turducken!

STEP 4

We know that $f(x) = x^2 + 3$ and $g(x) = 5x - 3$ .
So, $(f \circ g)(x)$ becomes $f(g(x))$ .
Let's substitute $g(x)$ into $f(x)$ : $f(g(x)) = (g(x))^2 + 3$

STEP 5

Now, let's replace $g(x)$ with its actual definition, which is $5x - 3$ : $(5x - 3)^2 + 3$

STEP 6

Time to expand that square!
Remember $(a - b)^2 = a^2 - 2ab + b^2$ .
Here, $a = 5x$ and $b = 3$ , so we get: $(5x)^2 - 2 \cdot (5x) \cdot 3 + 3^2 + 3$

STEP 7

Simplify that expression: $25x^2 - 30x + 9 + 3$ $25x^2 - 30x + 12$

STEP 8

Now that we have a nice and simplified expression for $(f \circ g)(x)$ , which is $25x^2 - 30x + 12$ , we can easily find $(f \circ g)(4)$ by substituting $x = 4$ .

STEP 9

Let's plug in $x = 4$ : $25 \cdot (4)^2 - 30 \cdot 4 + 12$

STEP 10

Calculate the square: $25 \cdot 16 - 30 \cdot 4 + 12$

STEP 11

Perform the multiplications: $400 - 120 + 12$

STEP 12

Combine the terms: $280 + 12$ $292$

STEP 13

a. $(f \circ g)(x) = 25x^2 - 30x + 12$ b. $(f \circ g)(4) = 292$

Was this helpful?

QuestionFor the given functions, $f(x) = x^2 + 3$ and $g(x) = 5x - 3$ , find the indicated composition. Write your answer by filling-in the blanks.
a. $(f \circ g)(x) =$
b. $(f \circ g)(4) =$
Moving to another question will save this response.
Question 21 of 23

Studdy Solution

STEP 1

What is this asking? We need to find the composition of two functions, $f(x)$ and $g(x)$ , both when $x$ is a variable and when $x$ is the number $4$ . Watch out! Function composition isn't just multiplying the functions!
It's about plugging one function into the other.

STEP 2

1. Find $(f \circ g)(x)$
2. Find $(f \circ g)(4)$

STEP 3

Alright, let's start with what $(f \circ g)(x)$ even means.
It means we take $g(x)$ and plug it in wherever we see an $x$ in $f(x)$ .
It's like a function turducken!

STEP 4

We know that $f(x) = x^2 + 3$ and $g(x) = 5x - 3$ .
So, $(f \circ g)(x)$ becomes $f(g(x))$ .
Let's substitute $g(x)$ into $f(x)$ : $f(g(x)) = (g(x))^2 + 3$

STEP 5

Now, let's replace $g(x)$ with its actual definition, which is $5x - 3$ : $(5x - 3)^2 + 3$

STEP 6

Time to expand that square!
Remember $(a - b)^2 = a^2 - 2ab + b^2$ .
Here, $a = 5x$ and $b = 3$ , so we get: $(5x)^2 - 2 \cdot (5x) \cdot 3 + 3^2 + 3$

STEP 7

Simplify that expression: $25x^2 - 30x + 9 + 3$ $25x^2 - 30x + 12$

STEP 8

Now that we have a nice and simplified expression for $(f \circ g)(x)$ , which is $25x^2 - 30x + 12$ , we can easily find $(f \circ g)(4)$ by substituting $x = 4$ .

STEP 9

Let's plug in $x = 4$ : $25 \cdot (4)^2 - 30 \cdot 4 + 12$

STEP 10

Calculate the square: $25 \cdot 16 - 30 \cdot 4 + 12$

STEP 11

Perform the multiplications: $400 - 120 + 12$

STEP 12

Combine the terms: $280 + 12$ $292$

STEP 13

a. $(f \circ g)(x) = 25x^2 - 30x + 12$ b. $(f \circ g)(4) = 292$

Get Studdy

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.

Studdy Solution

STEP 1

What is this asking? We need to find the composition of two functions, f(x)f(x)f(x) and g(x)g(x)g(x), both when xxx is a variable and when xxx is the number 444. Watch out! Function composition isn't just multiplying the functions!It's about plugging one function *into* the other.

STEP 2

1. Find (f∘g)(x)(f \circ g)(x)(f∘g)(x)2. Find (f∘g)(4)(f \circ g)(4)(f∘g)(4)

STEP 3

Alright, let's **start** with what (f∘g)(x)(f \circ g)(x)(f∘g)(x) even *means*.It means we take g(x)g(x)g(x) and plug it in wherever we see an xxx in f(x)f(x)f(x).It's like a function *turducken*!

STEP 4

We know that f(x)=x2+3f(x) = x^2 + 3f(x)=x2+3 and g(x)=5x−3g(x) = 5x - 3g(x)=5x−3.So, (f∘g)(x)(f \circ g)(x)(f∘g)(x) becomes f(g(x))f(g(x))f(g(x)).Let's **substitute** g(x)g(x)g(x) into f(x)f(x)f(x): f(g(x))=(g(x))2+3f(g(x)) = (g(x))^2 + 3f(g(x))=(g(x))2+3

STEP 5

Now, let's **replace** g(x)g(x)g(x) with its actual definition, which is 5x−35x - 35x−3: (5x−3)2+3(5x - 3)^2 + 3(5x−3)2+3

STEP 6

Time to **expand** that square!Remember (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2(a−b)2=a2−2ab+b2.Here, a=5xa = 5xa=5x and b=3b = 3b=3, so we get: (5x)2−2⋅(5x)⋅3+32+3(5x)^2 - 2 \cdot (5x) \cdot 3 + 3^2 + 3(5x)2−2⋅(5x)⋅3+32+3

STEP 7

**Simplify** that expression: 25x2−30x+9+325x^2 - 30x + 9 + 325x2−30x+9+3 25x2−30x+1225x^2 - 30x + 1225x2−30x+12

STEP 8

Now that we have a **nice and simplified** expression for (f∘g)(x)(f \circ g)(x)(f∘g)(x), which is 25x2−30x+1225x^2 - 30x + 1225x2−30x+12, we can **easily** find (f∘g)(4)(f \circ g)(4)(f∘g)(4) by **substituting** x=4x = 4x=4.

STEP 9

Let's **plug in** x=4x = 4x=4: 25⋅(4)2−30⋅4+1225 \cdot (4)^2 - 30 \cdot 4 + 1225⋅(4)2−30⋅4+12

STEP 10

**Calculate** the square: 25⋅16−30⋅4+1225 \cdot 16 - 30 \cdot 4 + 1225⋅16−30⋅4+12

STEP 11

**Perform** the multiplications: 400−120+12400 - 120 + 12400−120+12

STEP 12

**Combine** the terms: 280+12280 + 12280+12 292292292

STEP 13

a. (f∘g)(x)=25x2−30x+12(f \circ g)(x) = 25x^2 - 30x + 12(f∘g)(x)=25x2−30x+12 b. (f∘g)(4)=292(f \circ g)(4) = 292(f∘g)(4)=292

Get Studdy

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 find the composition of two functions, $f(x)$ and $g(x)$ , both when $x$ is a variable and when $x$ is the number $4$ . Watch out! Function composition isn't just multiplying the functions!
It's about plugging one function into the other.

1. Find $(f \circ g)(x)$
2. Find $(f \circ g)(4)$

Alright, let's start with what $(f \circ g)(x)$ even means.
It means we take $g(x)$ and plug it in wherever we see an $x$ in $f(x)$ .
It's like a function turducken!

We know that $f(x) = x^2 + 3$ and $g(x) = 5x - 3$ .
So, $(f \circ g)(x)$ becomes $f(g(x))$ .
Let's substitute $g(x)$ into $f(x)$ : $f(g(x)) = (g(x))^2 + 3$

Now, let's replace $g(x)$ with its actual definition, which is $5x - 3$ : $(5x - 3)^2 + 3$

Time to expand that square!
Remember $(a - b)^2 = a^2 - 2ab + b^2$ .
Here, $a = 5x$ and $b = 3$ , so we get: $(5x)^2 - 2 \cdot (5x) \cdot 3 + 3^2 + 3$

Simplify that expression: $25x^2 - 30x + 9 + 3$ $25x^2 - 30x + 12$

Now that we have a nice and simplified expression for $(f \circ g)(x)$ , which is $25x^2 - 30x + 12$ , we can easily find $(f \circ g)(4)$ by substituting $x = 4$ .

Let's plug in $x = 4$ : $25 \cdot (4)^2 - 30 \cdot 4 + 12$

Calculate the square: $25 \cdot 16 - 30 \cdot 4 + 12$

Perform the multiplications: $400 - 120 + 12$

Combine the terms: $280 + 12$ $292$

a. $(f \circ g)(x) = 25x^2 - 30x + 12$ b. $(f \circ g)(4) = 292$