Composite Functions

Given: f(x) = 2x + 3 and g(x) = 3x + M

If (fog)(x) = (gof)(x) Find M.

M = 

Step 1: Find (f∘g)(x)

The composite function (f∘g)(x) is equivalent to f(g(x)). We substitute the function g(x) into f(x).

2(3x + M) + 3

6x + 2M + 3

So, (f∘g)(x)=6x+2M+3.

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Step 2: Find (g∘f)(x)

The composite function (g∘f)(x) is equivalent to g(f(x)). We substitute the function f(x) into g(x).

3(2x + 3) + M

6x + 9 + M

So, (g∘f)(x)=6x+9+M.

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Step 3: Set the expressions equal and solve for M

The problem states that (f∘g)(x)=(g∘f)(x). We can now set the two expressions we found equal to each other and solve for the variable M.

6x+2M+3=6x+9+M

Subtract 6x from both sides of the equation:

2M+3=9+M

Subtract M from both sides:

M+3=9

Subtract 3 from both sides:

M=6

Given: f(x) = 9x + 3 and (f∘g)(x) = 81x + 57

Find g(x) = m x + n

m =

n =

Step 1: Find (f∘g)(x)

9(mx + n) + 3 = 81x  + 57

9mx + 9n + 3 = 81x + 57

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Step 2: Solve for m

9mx = 81x (Cross out x on both sides)

9m = 81 (Divide by 9 on both sides)

m = 9

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Step 3: Solve for n

9n + 3 = 57 (Subtract 3 on both sides)

9n = 54 (Divide by 9 on both sides)

n = 6

Given f(x) = 5 - 3mx and g(x) = nx - 6

Find m and n if (f∘g)(x) = -45x + 95

m =

n =

Step 1: Put g(x) into f(x)

5 - 3m (nx - 6) = -45x +95

5 - 3mnx + 18m = -45x +95

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Step 2:

-3mnx = -45x (Cross out x on both sides)

-3mn = -45

(Can’t solve yet because we don’t know m)

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Step 3: Solve for m

5 + 18m = 95 (Subtract 5 from both sides)

18m = 90 (Divide by 18 on both sides)

m = 5

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Step 4: Solve Step 2

-3mnx = -45x (Cross out x on both sides)

-3mn = -45

(-3)(5)n = -45 (We know m = -5 now)

-15n = -45 (Divide 15 from both sides)

 n = 3

Given: f(x) = 4x +4 and g(x) = 3x + 2

Evaluate: (4f - 6g)(-1) =

Step 1: Solve for f(x)

We know x = -1

4(-1)+4

f = 0

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Step 2: Solve for g(x)

3(-1) + 2 

g = -1

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Step 3: Evaluate (4f - 6g)(-1)

(4 ⋅ 0) - (6 ⋅ -1)

(0 - -6)

= 6

Given: f(x) = 5x + 8 and g(x) = 5x -5

Find: (f∘g)(2)

Step 1: Solve for g(x)

We know x = 2

5(2) - 5

= g(x) = 5

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Step 2: Put g(x) into f(x)

5(5) + 8

= 33

Given: f(x) = 7x + 3 and g(x) = 6x - 4

Find: f(2) + g(3) = 

Step 1: Solve for f(x)

We know that x is 2 in f(x)

7(2) + 3

= 17

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Step 2: Solve for g(x)

We know that x is 3 in g(x)

6(3) - 4

= 14

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Step 3: Add f(x) and g(x)

17 + 14

= 31

Given: f(x) = 2x + 8 and g(x) = 2x - 8

Find: f(2) - g(2)

Step 1: Solve for f(x)

We know that x is 2 in f(x)

2(2) + 8

= 12

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Step 2: Solve for g(x)

We know that x is 2 in g(x)

2(2) - 8

= -4

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Step 3: Subtract g(x) from f(x)

12 - - 4

=16

Given: f(x) = 7x + 7 and g(x) = 4x - 4

(f∘g)(a) = 175     a = 

Step 1: Put g(x) into f(x)

7(4x - 4) + 7

28x - 28 + 7

28x - 21

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Step 2: Solve for (f∘g)(a) = 175

28x - 21 = 175 (Move 21 to the other sider and make it positive)

28x = 196 (Subtract 28 from both sides)

x = 7

Given:  g(x - 7) =  5x - 7 ;   then   g(-2)  =

Step 1: Solve for x

We know that g = -2, so (x- 7) must equal -2

g(x - 7) = -2

x = 5

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Step 2: Solve for 5x - 7

We know that x is equal to 5

5(5) - 7

= 18

Given:

              5x- 5 ,   x < 1 

f(x)  =      x² - 9 ,    1 ≤ x ≤ 6     find (f ∘ f ∘ f)(6) =

              -x + 1 ,      x > 6

Step 1: Evaluate f(6)

First, we need to find the value of the innermost function, f(6). We look at the piecewise definition of f(x) to determine which part applies when x=6. The condition for the second piece is 1≤x≤6, which includes x=6.

f(x)= x²−9

Substituting x=6:

36 - 9

= 27

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Step 2: Evaluate f(f(6))

Now we use the result from Step 1, which is 27, as the input for the next function call. So we need to find f(27). We again look at the piecewise definition of f(x) to see which part applies when x=27. The condition for the third piece is x>6, which includes x=27.

f(x)=−x+1

Substituting x=27:

f(27)=−(27)+1

=−26

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Step 3: Evaluate f(f(f(6)))

Finally, we use the result from Step 2, which is -26, as the input for the final function call. So we need to find f(−26). We look at the piecewise definition of f(x) to see which part applies when x=−26. The condition for the first piece is x<1, which includes x=−26.

f(x)=5x−5

Substituting x=−26:

f(−26) = 5(−26) − 5 = −130 − 5

=−135

The value of (f∘f∘f)(6) is -135.

Given (f ∘ f)(x) =  9x + -56

and  f(-3) = 37   what is the general form of f(x)?

f(x) = x +

Step 1: Use the composite function to find the expression of f(f(x))

(We are given that f(x) = mx + b)

We are also given that (f∘f)(x)=9x−56. By comparing the coefficients of the x terms and the constant terms, we can create a system of equations.

m²x + (mb + b) = 9x − 56

Equating the x coefficients:

m²=9

Equating the constant terms:

mb + b =−56

(Since the square root of 9 is 3 and -3, m is one of these two)

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Step 2: Use the given point to find another relationship between m and b

We are given that f(−3) = 37. Using the general form of f(x) = mx + b, we can substitute these values.

f(−3) = m(−3) + b

37 = −3m + b

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Step 3: Solve the system of equations for m and b

Case 1: Let m = 3

Substitute m = 3 into the constant term equation from Step 1:

3b + b = -56

4b = -56

b = -14

Now, let's check if this pair of values (m=3,b=−14) works in the equation from Step 2:

37 = -3m + b

37 = -3(3) + (-14)

37 = -9 - 14

37 = −23

This is a false statement, so m != 3.

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Case 2: Let m=−3

Substitute m=−3 into the constant term equation from Step 1:

-3b + b = -56

-2b = -56

b = 28

Now, let's check if this pair of values (m=−3,b=28) works in the equation from Step 2:

37 = -3m + b

37 = -3(-3) + 28

37 = 9 + 28

37 = 37

This is a true statement, so the correct values are m=−3 and b=28

Given (f ∘ f)(x) =  4x - 24

and  f(-2) = -12   what is the general form of f(x)?

f(x) = x +

Step 1: Use the composite function to find the expression of f(f(x))

(We are given that f(x) = mx + b)

We are also given that (f∘f)(x) = 4x − 24. By comparing the coefficients of the x terms and the constant terms, we can create a system of equations.

m²x + (mb + b) = 4x − 24

Equating the x coefficients:

m² = 4

Equating the constant terms:

mb + b = -24

(Since the square root of 4 is 2 and -2, m is one of these two)

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Step 2: Use the given point to find another relationship between m and b

We are given that f(−2) = -12. Using the general form of f(x) = mx + b, we can substitute these values.

f(−2) = m(−2) + b

-12 = −2m + b

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Step 3: Solve the system of equations for m and b

Case 1: Let m = 2

Substitute m = 2 into the constant term equation from Step 1:

2b+b=-24

3b=-24

b = -8

Now, let's check if this pair of values (m=2,b=−8) works in the equation from Step 2:

-12=-2m+b

-12=-2(2)+(-8)

-12=-4-8

12 = 12

This is statement correct, so m = 2.

Given m(x) is a linear function. 

(m ∘ p)(x) = 28x² + 57 

and  p(x) = 4x² + 7

Find m(x) = x +

Step 1: Solve for m

(We know we (m ∘ p)(x) = 28x² +57)

p(x) = 4x² + 7

__(4x² + 7) +__

7(4x² + 7) + 8

= 28x² + 57

7x + 8

f(x) = 2x - a   x < 0    AND   5x - b  x ≥ 0 

g(x) = 8 x + 3 

If (g ∘ f)(-9) = -205  and 

(g ∘ f)(8) = 291  find a and b.

a =   and   b=

Step 1: Put x in the piecewise function and solve for f(x)

(We know that x is less than 0 for f, so we use the first piecewise function for f(x))

2x - a

2(-9) - a

-18 - a

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Step 2: Put f(x) into g(x)

8(-18 - a) + 3

-144 -8a + 3 = -205

-144 + 3 

=-141

-141 - 8a = -205 (Subtract -141 from both sides)

8a = 64 (Divide by 8 on both sides)

a = 8

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Step 3: Put x in the piecewise function and solve for f(x) for the second equation

(We know that x = 8 in this equation, so we use the second piecewise function for f(x))

5x - b

5(8) - b

40 - b

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Step 4: Put f(x) into g(x)

8(40 - b) + 3

320 - 8b + 3 = 291

320 + 3 = 323

323 - 8b = 291 (Subtract 323 from both sides)

-8b = -32 (Divide -8 from both sides) 

= 4

Given (f ∘ f)(x) =  4x - 12

Find f(x) = x +     

Step 1: Equate coefficients

(We know that mb + b = -12 and that m² is 4)

(So m can either be 2 or -2)

Case 1: m=2 Substitute m=2 into the second equation:

(2)b + b = -12

3b = -12

b = -4

This gives us the solution f(x)=2x−4.

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Case 2: m=−2 Substitute m=−2 into the second equation:

(-2)b + b = -12

-b = -12

b = 12

This gives us the solution f(x) = −2x + 12.

Since the problem does not provide any additional information to distinguish between these two cases, both are valid solutions.

Given m(x) = 3x + 8 and p(x) = cx² + d.

If  (p ∘ m)(x) =  54x² + 288x + 390,

then p(x) = x²

Step 1: Set up the composite function

The notation (p∘m)(x) means p(m(x)). We can substitute the expression for m(x) into p(x)=cx² + d.

Given: m(x) = 3x + 8 Given: p(x) = cx² + d

p(m(x)) = p(3x+8) p(3x+8) = c(3x+8)^2 + d

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Next, we expand the term (3x + 8)² using the formula for a perfect square trinomial, (a+b)² = a2 + 2ab + b²:

(3x+8)^2 = (3x)^2 + 2(3x)(8) + (8)^2(3x+8)^2 = 9x^2 + 48x + 64

Substitute this back into our expression for p(m(x)):

p(m(x)) = c(9x^2 + 48x + 64) + dp(m(x)) = 9cx^2 + 48cx + 64c + d

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Step 2: Equate coefficients and solve for c and d

We are given that (p∘m)(x)= 54 x² + 288x + 390. We can set our derived expression equal to this given expression and compare the coefficients of the corresponding terms.

9cx2+48cx+(64c+d)=54x2+288x+390

Solve for c: By equating the coefficients of the x2 terms on both sides, we get:

9c = 54c = \frac{54}{9}c = 6

Solve for d: By equating the constant terms on both sides, we get:

64c+d=390

Now, substitute the value of c=6 into this equation:

64(6) + d = 390384 + d = 390d = 390 - 384

d=6

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Step 3: Write the final function

With the values of c=6 and d=6, we can write the final form of the function p(x).

p(x)=cx2+d

p(x)=6x2+6

Given m(x) = 6x+7  and   (m ∘ p)(x) = 36x + 19  p(x) = x +

Step 1: Solve for p(x)

(mop) = 6() + 7 = 36x + 19

6(6x + 2) + 7

36x + 12 + 7

36x + 19

Given the following conditions determine the linear function, h(x).

a)    h(x) has the form h(x)=ax+b and a+b=9.

b)    n(x) is a constant function

c)    n[h(x)] = 8

d)   h[n(x)] = 37

h(x) = x +

Step 1: Use the constant function to find an equation for h(x)

Conditions (b) and (c) tell us that n(x) is a constant function and that n[h(x)]=8. This means the output of the function n(x) is always 8, regardless of the input. So, we can determine that n(x)=8.

Next, we use condition (d), which states that h[n(x)]=37. We can substitute n(x)=8 into this equation:

h(8)=37

Since we know h(x) is a linear function of the form h(x)=ax+b, we can now write an equation using this point (8,37):

a(8)+b=37

8a+b=37

This is our second equation.

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Step 2: Solve the system of linear equations

We now have a system of two linear equations from condition (a) and our result from Step 1.

1. a+b=9

2. 8a+b=37

To solve for a and b, we can subtract the first equation from the second to eliminate b.

(8a + b) - (a + b) = 37 - 97a = 28a = 4

Now, substitute the value of a back into the first equation to solve for b:

4 + b = 9b = 5

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Step 3: Write the final function

With the values a=4 and b=5, the function h(x) is:

h(x)=4x+5

Given: f(x-3) = 3x - 3  and  g(x-6) = 5x -3

Find: (f ∘ g)(4)  =  

Step 1: Find g(4)

The function is defined as g(x−6)=5x−3. To find g(4), we must determine the value of x that makes the input, (x−6), equal to 4.

x−6=4 x=10

Now, substitute this value of x=10 into the expression 5x−3:

g(4)=5(10)−3=50−3=47

So, g(4)=47.

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Step 2: Find f(g(4))

Since we found that g(4)=47, we now need to find f(47). The function is defined as f(x−3)=3x−3. To find f(47), we must determine the value of x that makes the input, (x−3), equal to 47.

x−3=47 x=50

Now, substitute this value of x=50 into the expression 3x−3:

f(47)=3(50)−3=150−3=147

Therefore, (f∘g)(4)=147.