# The function f that gives the income tax on income x .

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.7, Problem 87E

(a)

To determine

## The function f that gives the income tax on income x.

Expert Solution

The function f for the income tax is f(x)={0.1x,0x200002000+0.2(x20000),x>20000 .

### Explanation of Solution

Given:

The income tax on income less than €20000 is 10% and the tax on income above €20000 is €20000 and 20% of the income.

Calculation:

Let the income tax be x,

Tax for the income €20000 is,

f(x)=10%ofincome=0.1x (1)

Tax for the income more than €20000 is,

f(x)=20000+20%oftheincome=20000+0.2(x20000) (2)

From the equation (1) and (2),

f(x)={0.1x0x200002000+0.2(x20000)x>20000

Thus, the income tax for income x is f(x)={0.1x0x200002000+0.2(x20000)x>20000

(b)

To determine

### The inverse of the function f(x)={0.1x                                  0≤x≤200002000+0.2(x−20000)             x>20000 .

Expert Solution

The inverse of the function f(x)={0.1x0x200002000+0.2(x20000)x>20000 is f1(x)={10x0x200005x+10000x>20000

### Explanation of Solution

Given:

The income tax on income less than €20000 is 10% and the tax on income above €20000 is €20000 and 20% of the income. The function is f(x)={0.1x0x200002000+0.2(x20000)x>20000

Calculation:

Solve for the income less than €20000,

Let the function be,

y=0.1x

Solve for the value of x,

10y=x

Interchange the value of x with y,

y=10x (1)

Thus, the value of inverse of the function is f1(x)=10x .

Solve for the income more than €20000,

Let the function be,

y=2000+0.2(x20000)

Solve for the value of x,

y2000=0.2(x20000)y2000=0.2x4000y2000+4000=0.2x

Further solve the value,

y+2000=0.2xy+20000.2=x5y+10000=x

Interchange the value of x with y,

y=5x+10000 (2)

Thus, the value of inverse of the function is f1(x)=5x+10000 .

The inverse of the function f(x)={0.1x0x200002000+0.2(x20000)x>20000 is f1(x)={10x0x200005x+10000x>20000 The function represents the rate of income tax on income.

(c)

To determine

### The income for the income tax €10000.

Expert Solution

The income for the income tax is 60000.

### Explanation of Solution

Given:

The value of inverse of the function is f1(x)=5x+10000 .

Calculation:

The formula to calculate the exchange of money is,

f1(x)=5x+10000 (1)

Substitute €10000 for x in equation (1),

f1(10000)=5×10000+10000=60000

Thus income for income tax is 60000.

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