Answer :
To determine the domain and range of the function [tex]\( f(x) = x^4 - 2x^2 - 4 \)[/tex], we need to analyze the behavior of the function.
### Step 1: Determining the Domain
The function [tex]\( f(x) = x^4 - 2x^2 - 4 \)[/tex] is a polynomial function. Polynomial functions are defined for all real numbers since there are no restrictions (like division by zero or taking the square root of a negative number) that limit the domain.
Domain: All real numbers
### Step 2: Determining the Range
To find the range, we need to analyze where the function [tex]\( f(x) \)[/tex] takes its minimum and maximum values.
### Step 2.1: Finding Critical Points
Critical points occur where the first derivative [tex]\( f'(x) \)[/tex] is zero or undefined. Let's find [tex]\( f'(x) \)[/tex]:
[tex]\[ f(x) = x^4 - 2x^2 - 4 \][/tex]
[tex]\[ f'(x) = 4x^3 - 4x \][/tex]
Set the first derivative equal to zero to find critical points:
[tex]\[ 4x^3 - 4x = 0 \][/tex]
[tex]\[ 4x(x^2 - 1) = 0 \][/tex]
[tex]\[ 4x(x - 1)(x + 1) = 0 \][/tex]
The critical points are:
[tex]\[ x = 0, 1, -1 \][/tex]
### Step 2.2: Evaluating the Function at Critical Points
Next, evaluate the function [tex]\( f(x) \)[/tex] at these critical points:
[tex]\[ f(0) = (0)^4 - 2(0)^2 - 4 = -4 \][/tex]
[tex]\[ f(1) = (1)^4 - 2(1)^2 - 4 = 1 - 2 - 4 = -5 \][/tex]
[tex]\[ f(-1) = (-1)^4 - 2(-1)^2 - 4 = 1 - 2 - 4 = -5 \][/tex]
### Step 2.3: Evaluating Limits as [tex]\( x \)[/tex] Approaches Infinity
Since [tex]\( f(x) = x^4 - 2x^2 - 4 \)[/tex] is a polynomial with a leading term [tex]\( x^4 \)[/tex], as [tex]\( x \)[/tex] approaches infinity or negative infinity, the [tex]\( x^4 \)[/tex] term dominates:
[tex]\[ \lim_{x \to \infty} f(x) = \infty \][/tex]
[tex]\[ \lim_{x \to -\infty} f(x) = \infty \][/tex]
From these evaluations, we see that the lowest function value occurs at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex], where [tex]\( f(1) = f(-1) = -5 \)[/tex].
### Step 2.4: Conclusion on Range
Since [tex]\( f(x) \)[/tex] has a minimum value of [tex]\(-5\)[/tex] and goes to infinity as [tex]\( x \)[/tex] goes to positive or negative infinity, the range is all real numbers greater than or equal to [tex]\(-5\)[/tex].
Range: All real numbers greater than or equal to [tex]\(-5\)[/tex]
### Final Answer
The function [tex]\( f(x) = x^4 - 2x^2 - 4 \)[/tex] has:
- Domain: All real numbers
- Range: All real numbers greater than or equal to [tex]\(-5\)[/tex]
### Step 1: Determining the Domain
The function [tex]\( f(x) = x^4 - 2x^2 - 4 \)[/tex] is a polynomial function. Polynomial functions are defined for all real numbers since there are no restrictions (like division by zero or taking the square root of a negative number) that limit the domain.
Domain: All real numbers
### Step 2: Determining the Range
To find the range, we need to analyze where the function [tex]\( f(x) \)[/tex] takes its minimum and maximum values.
### Step 2.1: Finding Critical Points
Critical points occur where the first derivative [tex]\( f'(x) \)[/tex] is zero or undefined. Let's find [tex]\( f'(x) \)[/tex]:
[tex]\[ f(x) = x^4 - 2x^2 - 4 \][/tex]
[tex]\[ f'(x) = 4x^3 - 4x \][/tex]
Set the first derivative equal to zero to find critical points:
[tex]\[ 4x^3 - 4x = 0 \][/tex]
[tex]\[ 4x(x^2 - 1) = 0 \][/tex]
[tex]\[ 4x(x - 1)(x + 1) = 0 \][/tex]
The critical points are:
[tex]\[ x = 0, 1, -1 \][/tex]
### Step 2.2: Evaluating the Function at Critical Points
Next, evaluate the function [tex]\( f(x) \)[/tex] at these critical points:
[tex]\[ f(0) = (0)^4 - 2(0)^2 - 4 = -4 \][/tex]
[tex]\[ f(1) = (1)^4 - 2(1)^2 - 4 = 1 - 2 - 4 = -5 \][/tex]
[tex]\[ f(-1) = (-1)^4 - 2(-1)^2 - 4 = 1 - 2 - 4 = -5 \][/tex]
### Step 2.3: Evaluating Limits as [tex]\( x \)[/tex] Approaches Infinity
Since [tex]\( f(x) = x^4 - 2x^2 - 4 \)[/tex] is a polynomial with a leading term [tex]\( x^4 \)[/tex], as [tex]\( x \)[/tex] approaches infinity or negative infinity, the [tex]\( x^4 \)[/tex] term dominates:
[tex]\[ \lim_{x \to \infty} f(x) = \infty \][/tex]
[tex]\[ \lim_{x \to -\infty} f(x) = \infty \][/tex]
From these evaluations, we see that the lowest function value occurs at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex], where [tex]\( f(1) = f(-1) = -5 \)[/tex].
### Step 2.4: Conclusion on Range
Since [tex]\( f(x) \)[/tex] has a minimum value of [tex]\(-5\)[/tex] and goes to infinity as [tex]\( x \)[/tex] goes to positive or negative infinity, the range is all real numbers greater than or equal to [tex]\(-5\)[/tex].
Range: All real numbers greater than or equal to [tex]\(-5\)[/tex]
### Final Answer
The function [tex]\( f(x) = x^4 - 2x^2 - 4 \)[/tex] has:
- Domain: All real numbers
- Range: All real numbers greater than or equal to [tex]\(-5\)[/tex]
