SOLUTIONS OF THE FUNCTION QUESTIONS

High school mathematics lesson. To find the composite of functions. Inverse functions. Operations on functions. The functions test-1 solutions.


Solution 1

f(2x + 10) = 6x + 14

We must do x the 2x + 1 expression for to find the f(x), from the f(2x + 1).

We must take the inverse of this expression for to do x the 2x + 1 expression.

y = 2x + 10

Let we write x instead of y, y instead of x.

x = 2y + 10

x – 10 = 2y

y = x – 10
2




If we write this expression instead of all x in the function f(2x + 10) , we finds the function f(x).

f(2•x – 10 
+10) = 6•x – 10) + 14
2




f(x – 10 + 10) = 3x – 30 + 14

f(x) = 3x – 16

The right answer is option B


Solution 2 

f(x) = 12x – 36 

We can write this function in the following form.

y = 12x – 36

We will write x instead of y, y instead of x.

x = 12y – 36

x + 36 = 12y

y = x + 36
12




f-1(x) =x + 36
12




The right answer is the option C.


Solution 3 

We can write the function f(x) = 16x + 24 as the following.

y = 16x + 24

Let we write x instead of y, y instead of x.

x = 16y + 24

x – 24 = 16y

y = x – 24
16




f-1(x) = x – 24
16




The right answer is option A.


Solution 4

f(x) = 12x - 8
5



y = 12x - 8
5



x = 12y - 8
5




5x = 12y – 8

5x + 8 = 12y

y = 5x + 8
12




The right answer is option E.


Solution 5

The inverse of inverse of a function is equal to it. 

[f(x)-1]-1 = f(x)

f-1(x) = x + 9
4




If we take inverse of this function, we get the function f(x).

Suppose, f-1 (x) = y .

y = x + 9
4



Let we write x instead of y and y instead of x.

x = y + 9
4




4x = y + 9

4x – 9 = y

f(x) = 4x – 9 

x = 5 için,

f(5) = 4•5 – 9

f(5) = 11


The right answer is option D.


Solution 6 

f(x) = 6x2 + 4

g(x) = 5x + 3

[f + g](x) = f(x) + g(x)

[f + g](x) = 6x2 + 4 + 5x + 3

[f + g](x) = 6x2 + 5x + 7

[f + g](3) = 6•32 + 5•3 + 7

[f + g](3) = 76


[f – g](x) = f(x) – g(x)

[f – g](x) = 6x2 + 4 – 5x – 3

[f – g](x) = 6x2 – 5x + 1

[f – g](1) = 6 – 5 + 2

[f – g](1) = 2



[f + g](3) 
 = 76
[f – g](1) 




= 38


The right answer is option D


Solution 7

To find the composition of the f(x) and g(x) functions, we write the function g(x) instead of x that in the function f(x).

f(x) = 3x + 24

g(x) = 5x – 8

fog(x) = 3(5x – 8) + 24

fog(x) = 15x – 24 + 24

fog(x) = 15x


The right answer is option C


Solution 8 

f(x) = x2 – 4x

g(x) = x + 2

We must find the function fog(x), firstly. Then, we takes inverse of this composite function.

We can compute the desired value by typing x = 21, in the inverse function.

fog(x) = (x + 2)2 – 4(x + 2)

fog(x) = x2 + 4x + 4 – 4x – 8 

fog(x) = x2 – 4

Now, let we find inverse of this function.

y = x2 – 4

x →y 

x = y2 - 4 

y2 = x + 4

y = √x + 4 


x = 21 için,

y = √4 + 21 

y = 5


The right answer is the option B


Solution 9 

[gof](x + 6) = 5x + 11

f-1 (x) = x + 1

Firstly, let we find gof(x) function. Then, we can find function f(x). Finally, we can find function g(x) from gof(x) using the function f(x).

y = x + 6

x = y + 6

y = x – 6

If we write x – 6 instead of x in the function gof(x + 6), we get the function gof(x).

gof(x – 6 + 6) = 5(x – 6) + 11

gof(x) = 5x – 19

Now, let we find the function f(x). The inverse of inverse of a function is equal to it. To find the function f(x), we must find inverse of the function f-1 (x).

f-1 (x) = x + 1

y = x + 1

x = y + 1

y = x – 1

f(x) = x – 1 


gof(x) = 5x – 19

g(f(x)) = 5x – 19

g(x – 1) = 5x – 19

To get the function g(x), we must take inverse of the x – 1 expression. Inverse of (x – 1) is x + 1. We write this expression instead of x in the function g(x – 1).

g(x + 1 – 1) = 5(x + 1) – 19

g(x) = 5x + 5 – 19

g(x) = 5x – 14

For x = 5,

g(5) = 5•5 – 14

= 11


The right answer is option B



Solution 10

f(x) =5x+2
3x+16




y =5x+2
3x+16




Let we write x, instead of y, y instead of x.

x =5y+2 +
3y+16



3xy + 16x = 5y + 2

5y – 3xy = 16x – 2

y(5 – 3x) = 16x – 2 

y =16x – 2
5 – 3x




f-1(x) =16x – 2
5 – 3x




x = 3 için,

f-1(2) = 16•2 -2
5-3•2




f-1 (2) = - 30


The right answer is option E


Solution 11 

fog(x) = 6x2 – 4

f(x) = 2x + 6

fog(x) = f(g(x))

f(g(x)) = 2g(x) + 6

6x2 – 4 = 2g(x) + 6

6x2 – 10 = 2g(x)

g(x) = 3x2 – 5


The right answer is option C


Solution 12

f(x + g(x)) = x2 + 6

Also,

g(2) = 12

for x = 2,

f(2 + g(2)) = 22 + 6

f(2 + 12) = 4 + 6

f(14) = 10


f-1(10) = 14


The right answer is option E


The Questions

Inverse Functions

Compound Functions




RISE KNOWLEDGE

07/10/2018

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