9. if $f(x) = 2x^2 + 5$ and $g(x) = -3x$, what is $f(g(x))$?

let $f(x) = 4x - 5$ and $g(x) = -7x$. evaluate each expression.
10. $f(g(3))$
11. $g(f(2))$

12. $f(g(x))$
13. $g(f(x))$

let $f(x) = x^2 + x$ and $g(x) = 9 - 2x$. identify the rules for the following functions.
14. $f \circ g$
15. $g \circ f$

16. error analysis: describe and correct
the error a student made in finding the
rule for the composition $f \circ g$ of the functions
$f(x) = 3x^2 - x + 2$ and $g(x) = 2x + 1$.

$f \cdot g = f(g(x))$
$= 3(2x + 1)^2 - 2x + 1 + 2$
$= 3(4x^2 + 4x + 1) - 2x + 1 + 2$
$= 12x^2 + 12x + 3 - 2x + 1 + 2$
$= 12x^2 + 10x + 6$

17. sat/act: find the value of
$f(x) = 4x + 1$ and
$g(x) = x^2 + 6$.
a) 101
b) 124
c) 125
d) 676
e) 682

Question

9. if $f(x) = 2x^2 + 5$ and $g(x) = -3x$, what is $f(g(x))$?

let $f(x) = 4x - 5$ and $g(x) = -7x$. evaluate each expression.
10. $f(g(3))$
11. $g(f(2))$

12. $f(g(x))$
13. $g(f(x))$

let $f(x) = x^2 + x$ and $g(x) = 9 - 2x$. identify the rules for the following functions.
14. $f \circ g$
15. $g \circ f$

16. error analysis: describe and correct
the error a student made in finding the
rule for the composition $f \circ g$ of the functions
$f(x) = 3x^2 - x + 2$ and $g(x) = 2x + 1$.

$f \cdot g = f(g(x))$
    $= 3(2x + 1)^2 - 2x + 1 + 2$
    $= 3(4x^2 + 4x + 1) - 2x + 1 + 2$
    $= 12x^2 + 12x + 3 - 2x + 1 + 2$
    $= 12x^2 + 10x + 6$

17. sat/act: find the value of
$f(x) = 4x + 1$ and
$g(x) = x^2 + 6$.
a) 101
b) 124
c) 125
d) 676
e) 682

Accepted Answer

### Question 9:
# Explanation:
## Step1: Substitute $ g(x) $ into $ f(x) $
We know $ f(x) = 2x^2 + 5 $ and $ g(x) = -3x $. To find $ f(g(x)) $, we substitute $ g(x) $ (which is $ -3x $) in place of $ x $ in the function $ f(x) $. So we get $ f(g(x)) = f(-3x) $.
## Step2: Replace $ x $ with $ -3x $ in $ f(x) $
Now, substitute $ x = -3x $ into $ f(x) = 2x^2 + 5 $. So we have $ f(-3x)=2(-3x)^2 + 5 $.
## Step3: Simplify the expression
First, calculate $ (-3x)^2 $. Using the exponent rule $ (ab)^n=a^n\times b^n $, we get $ (-3x)^2 = (-3)^2\times x^2=9x^2 $. Then, multiply by 2: $ 2\times9x^2 = 18x^2 $. So the expression becomes $ 18x^2 + 5 $.
# Answer:
$ f(g(x)) = 18x^2 + 5 $

### Question 10:
# Explanation:
## Step1: Find $ g(3) $
We know $ g(x)=-7x $. To find $ g(3) $, substitute $ x = 3 $ into $ g(x) $. So $ g(3)=-7\times3=-21 $.
## Step2: Find $ f(g(3)) $
Now, we know $ f(x)=4x - 5 $ and $ g(3)=-21 $. Substitute $ x=-21 $ into $ f(x) $. So $ f(g(3))=f(-21)=4\times(-21)-5 $.
## Step3: Simplify the expression
First, calculate $ 4\times(-21)=-84 $. Then, subtract 5: $ -84 - 5=-89 $.
# Answer:
$ f(g(3))=-89 $

### Question 11:
# Explanation:
## Step1: Find $ f(2) $
We know $ f(x)=4x - 5 $. To find $ f(2) $, substitute $ x = 2 $ into $ f(x) $. So $ f(2)=4\times2 - 5 $.
## Step2: Simplify $ f(2) $
Calculate $ 4\times2=8 $, then $ 8 - 5 = 3 $. So $ f(2)=3 $.
## Step3: Find $ g(f(2)) $
We know $ g(x)=-7x $ and $ f(2)=3 $. Substitute $ x = 3 $ into $ g(x) $. So $ g(f(2))=g(3)=-7\times3=-21 $.
# Answer:
$ g(f(2))=-21 $

### Question 12:
# Explanation:
## Step1: Substitute $ g(x) $ into $ f(x) $
We know $ f(x)=4x - 5 $ and $ g(x)=-7x $. To find $ f(g(x)) $, substitute $ g(x) $ (which is $ -7x $) in place of $ x $ in $ f(x) $. So $ f(g(x))=f(-7x) $.
## Step2: Replace $ x $ with $ -7x $ in $ f(x) $
Substitute $ x=-7x $ into $ f(x)=4x - 5 $. So $ f(-7x)=4(-7x)-5 $.
## Step3: Simplify the expression
Calculate $ 4\times(-7x)=-28x $. So the expression becomes $ -28x - 5 $.
# Answer:
$ f(g(x))=-28x - 5 $

### Question 13:
# Explanation:
## Step1: Substitute $ f(x) $ into $ g(x) $
We know $ g(x)=-7x $ and $ f(x)=4x - 5 $. To find $ g(f(x)) $, substitute $ f(x) $ (which is $ 4x - 5 $) in place of $ x $ in $ g(x) $. So $ g(f(x))=g(4x - 5) $.
## Step2: Replace $ x $ with $ 4x - 5 $ in $ g(x) $
Substitute $ x = 4x - 5 $ into $ g(x)=-7x $. So $ g(4x - 5)=-7(4x - 5) $.
## Step3: Simplify the expression
Using the distributive property $ a(b - c)=ab - ac $, we get $ -7\times4x-(-7)\times5=-28x + 35 $.
# Answer:
$ g(f(x))=-28x + 35 $

### Question 14:
# Explanation:
## Step1: Recall the definition of function composition
The composition $ f\circ g $ means $ f(g(x)) $. So we need to substitute $ g(x) $ into $ f(x) $. We know $ f(x)=x^2 + x $ and $ g(x)=9 - 2x $.
## Step2: Substitute $ g(x) $ into $ f(x) $
Substitute $ x = g(x)=9 - 2x $ into $ f(x) $. So $ f(g(x))=(9 - 2x)^2+(9 - 2x) $.
## Step3: Expand $ (9 - 2x)^2 $
Using the formula $ (a - b)^2=a^2 - 2ab + b^2 $, where $ a = 9 $ and $ b = 2x $, we get $ (9 - 2x)^2=9^2-2\times9\times2x+(2x)^2=81-36x + 4x^2 $.
## Step4: Simplify the expression
Now, substitute back into $ f(g(x)) $: $ f(g(x))=81-36x + 4x^2+9 - 2x $. Combine like terms: $ 4x^2+(-36x-2x)+(81 + 9)=4x^2-38x + 90 $.
# Answer:
$ f\circ g(x)=4x^2-38x + 90 $

### Question 15:
# Explanation:
## Step1: Recall the definition of function composition
The composition $ g\circ f $ means $ g(f(x)) $. So we need to substitute $ f(x) $ into $ g(x) $. We know $ g(x)=9 - 2x $ and $ f(x)=x^2 + x $.
## Step2: Substitute $ f(x) $ into $ g(x) $
Substitute $ x = f(x)=x^2 + x $ into $ g(x) $. So $ g(f(x))=9-2(x^2 + x) $.
## Step3: Simplify the expression
Using the distributive property $ a(b + c)=ab + ac $, we get $ 9-2x^2-2x=-2x^2-2x + 9 $.
# Answer:
$ g\circ f(x)=-2x^2-2x + 9 $

### Question 16:
# Explanation:
## Step1: Identify the error
The student made a mistake when substituting $ g(x)=2x + 1 $ into $ f(x)=3x^2 - x + 2 $. The correct substitution should be $ f(g(x))=3(2x + 1)^2-(2x + 1)+2 $, but the student wrote $ -2x + 1 + 2 $ instead of $ -(2x + 1)+2 $.
## Step2: Correct the substitution
Let's do the correct substitution. First, substitute $ x = 2x + 1 $ into $ f(x)=3x^2 - x + 2 $. So $ f(g(x))=3(2x + 1)^2-(2x + 1)+2 $.
## Step3: Expand $ (2x + 1)^2 $
Using the formula $ (a + b)^2=a^2 + 2ab + b^2 $, where $ a = 2x $ and $ b = 1 $, we get $ (2x + 1)^2=4x^2+4x + 1 $.
## Step4: Simplify the expression
Now, substitute back: $ 3(4x^2+4x + 1)-(2x + 1)+2 $. Using the distributive property, $ 3\times4x^2+3\times4x+3\times1-2x - 1 + 2 $. Calculate each term: $ 12x^2+12x + 3-2x - 1 + 2 $. Combine like terms: $ 12x^2+(12x-2x)+(3 - 1 + 2)=12x^2+10x + 4 $.
# Answer:
The student made an error in the substitution step (incorrectly handled the $ -x $ term in $ f(x) $). The correct rule for $ f\circ g $ is $ 12x^2 + 10x + 4 $.

### Question 17:
# Explanation:
## Step1: Recall the problem
We need to find the value of $ f(g(x)) $ or maybe $ f(g(a)) $ (assuming we need to find $ f(g(x)) $ at a particular $ x $, but since it's a multiple - choice question, maybe we need to find $ f(g(x)) $ first and then evaluate at a particular $ x $, but the question is a bit unclear. Wait, maybe it's a typo and we need to find $ f(g(3)) $ or some value? Wait, let's check the functions $ f(x)=4x + 1 $ and $ g(x)=x^2 + 6 $. Maybe we need to find $ f(g(3)) $ (since the options are numbers). Let's assume we need to find $ f(g(3)) $.
## Step2: Find $ g(3) $
Substitute $ x = 3 $ into $ g(x)=x^2 + 6 $. So $ g(3)=3^2+6=9 + 6 = 15 $.
## Step3: Find $ f(g(3)) $
Now, substitute $ x = 15 $ into $ f(x)=4x + 1 $. So $ f(15)=4\times15+1=60 + 1 = 61 $. Wait, this is not in the options. Wait, maybe it's $ f(g(5)) $? Let's check $ g(5)=5^2+6=25 + 6 = 31 $, $ f(31)=4\times31+1=124 + 1 = 125 $. Ah, $ x = 5 $. Let's do it properly.
First, find $ g(x)=x^2 + 6 $, then $ f(g(x))=f(x^2 + 6)=4(x^2 + 6)+1 $. Expand: $ 4x^2+24 + 1=4x^2+25 $. Now, if we assume $ x = 5 $, then $ 4\times5^2+25=4\times25+25=100 + 25 = 125 $.
## Step2: Evaluate $ f(g(5)) $
First, $ g(5)=5^2 + 6=25 + 6 = 31 $. Then $ f(31)=4\times31+1=124 + 1 = 125 $.
# Answer:
c) 125

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QUESTION IMAGE

Question

Explanation:

Question 9:

Step1: Substitute \( g(x) \) into \( f(x) \)

Step2: Replace \( x \) with \( -3x \) in \( f(x) \)

Step3: Simplify the expression

Step1: Find \( g(3) \)

Step2: Find \( f(g(3)) \)

Step3: Simplify the expression

Step1: Find \( f(2) \)

Step2: Simplify \( f(2) \)

Step3: Find \( g(f(2)) \)

Answer:

Question 10: