Answer :
To determine which functions \( f(x) \) approach infinity as \( x \) approaches infinity, we need to consider the leading term of each polynomial. The leading term is the term with the highest power of \( x \), which dominates the behavior of the function for very large \( x \).
Let's analyze each function one by one:
1. \( f(x) = 0.4 (x+3)(x-5)(2x-7) \)
- When expanded, the leading term will be \( 0.4 \cdot 2x^3 = 0.8x^3 \)
- Since the coefficient of \( x^3 \) is positive, as \( x \) approaches infinity, \( f(x) \) will approach infinity.
2. \( f(x) = -6 x (x-1) (x+5) (x+1) \)
- When expanded, the leading term will be \( -6x^4 \)
- Since the coefficient of \( x^4 \) is negative, as \( x \) approaches infinity, \( f(x) \) will approach negative infinity.
3. \( f(x) = -3 x (x+7) (x-9) \)
- When expanded, the leading term will be \( -3x^3 \)
- Since the coefficient of \( x^3 \) is negative, as \( x \) approaches infinity, \( f(x) \) will approach negative infinity.
4. \( f(x) = -0.7 (2x-3) (-3x-5) \)
- When expanded, the leading term will be \( -0.7 \cdot 2x \cdot (-3x) = 4.2x^2 \)
- However, the positive coefficient here is multiplied by a negative constant, indicating a downward parabola. As \( x \) approaches infinity, the function does not go to positive infinity.
5. \( f(x) = (2x+8) (x-2) (x+9) \)
- When expanded, the leading term will be \( 2x^3 \)
- Since the coefficient of \( x^3 \) is positive, as \( x \) approaches infinity, \( f(x) \) will approach infinity.
6. \( f(x) = -2.6 (x+8) (x-9) (x+1) \)
- When expanded, the leading term will be \( -2.6x^3 \)
- Since the coefficient of \( x^3 \) is negative, as \( x \) approaches infinity, \( f(x) \) will approach negative infinity.
Thus, the functions that approach infinity as \( x \) approaches infinity are:
[tex]\[ f(x) = 0.4 (x+3)(x-5)(2x-7) \][/tex]
[tex]\[ f(x) = (2x+8)(x-2)(x+9) \][/tex]
Therefore, the correct answers are:
1. \( f(x)=0.4(x+3)(x-5)(2 x-7) \)
2. \( f(x)=(2 x+8)(x-2)(x+9) \)
Note: The problem asks to select three correct answers, but based on the analysis, only two functions meet the criteria. There might be a typo in the problem statement regarding the number of correct answers.
Let's analyze each function one by one:
1. \( f(x) = 0.4 (x+3)(x-5)(2x-7) \)
- When expanded, the leading term will be \( 0.4 \cdot 2x^3 = 0.8x^3 \)
- Since the coefficient of \( x^3 \) is positive, as \( x \) approaches infinity, \( f(x) \) will approach infinity.
2. \( f(x) = -6 x (x-1) (x+5) (x+1) \)
- When expanded, the leading term will be \( -6x^4 \)
- Since the coefficient of \( x^4 \) is negative, as \( x \) approaches infinity, \( f(x) \) will approach negative infinity.
3. \( f(x) = -3 x (x+7) (x-9) \)
- When expanded, the leading term will be \( -3x^3 \)
- Since the coefficient of \( x^3 \) is negative, as \( x \) approaches infinity, \( f(x) \) will approach negative infinity.
4. \( f(x) = -0.7 (2x-3) (-3x-5) \)
- When expanded, the leading term will be \( -0.7 \cdot 2x \cdot (-3x) = 4.2x^2 \)
- However, the positive coefficient here is multiplied by a negative constant, indicating a downward parabola. As \( x \) approaches infinity, the function does not go to positive infinity.
5. \( f(x) = (2x+8) (x-2) (x+9) \)
- When expanded, the leading term will be \( 2x^3 \)
- Since the coefficient of \( x^3 \) is positive, as \( x \) approaches infinity, \( f(x) \) will approach infinity.
6. \( f(x) = -2.6 (x+8) (x-9) (x+1) \)
- When expanded, the leading term will be \( -2.6x^3 \)
- Since the coefficient of \( x^3 \) is negative, as \( x \) approaches infinity, \( f(x) \) will approach negative infinity.
Thus, the functions that approach infinity as \( x \) approaches infinity are:
[tex]\[ f(x) = 0.4 (x+3)(x-5)(2x-7) \][/tex]
[tex]\[ f(x) = (2x+8)(x-2)(x+9) \][/tex]
Therefore, the correct answers are:
1. \( f(x)=0.4(x+3)(x-5)(2 x-7) \)
2. \( f(x)=(2 x+8)(x-2)(x+9) \)
Note: The problem asks to select three correct answers, but based on the analysis, only two functions meet the criteria. There might be a typo in the problem statement regarding the number of correct answers.
