# The domain of the function f ( x ) = 2 x + 1 x 2 + x − 2 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter T, Problem 3CDT

(a)

To determine

## To find: The domain of the function f(x)=2x+1x2+x−2.

Expert Solution

The domain of the function f(x)=2x+1x2+x2 is (,2)(2,1)(1,).

### Explanation of Solution

Formula used:

Domain: The set of all possible values of the independent variables in a function is called the domain of the function.

Calculation:

Consider the given function f(x)=2x+1x2+x2.

The function is not defined if the denominator of the function equals to 0.

The denominator of the function f(x)=2x+1x2+x2 is x2+x2.

Thus,

x2+x2=0(x+2)(x1)=0

Therefore, the function f(x)=2x+1x2+x2 is not defined for the values x=2and x=1.

Hence, from the definition stated above, it is clear that the domain of the function is (,2)(2,1)(1,).

Thus, the domain of the function f(x)=2x+1x2+x2 is (,2)(2,1)(1,).

(b)

To determine

### To find: The domain of the function g(x)=x3x2+1.

Expert Solution

The domain of the function g(x)=x3x2+1 is (,).

### Explanation of Solution

Formula used:

Domain: The set of all possible values of the independent variables in a function is called the domain of the function.

Calculation:

Consider the given function g(x)=x3x2+1.

The function is not defined if the denominator of the function equals to 0.

The denominator of the function g(x)=x3x2+1 is x2+1.

The denominator is x2+1 never zero for all the real values.

It is clear that, the function g(x)=x3x2+1 is defined for all the real values.

Hence, from the definition stated above, it is clear that the domain of the function is (,).

Thus, the domain of the function g(x)=x3x2+1 is (,).

(c)

To determine

### To find: The domain of the function h(x)=4−x+x2−1.

Expert Solution

The domain of the function h(x)=4x+x21 is (,1][1,4].

### Explanation of Solution

Formula used:

Domain: The set of all possible values of the independent variables in a function is called the domain of the function.

Calculation:

Consider the given function h(x)=4x+x21.

The function contains terms that is inside a square root.

It is known that the function is defined if the expression under the square root is non-negative.

That is,

4x0x4

And,

x210(x+1)(x1)0

Thus, x1 or x1.

Hence, from the definition stated above, it is clear that the domain of the function is (,1][1,4].

Thus, the domain of the function h(x)=4x+x21 is (,1][1,4].

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