The end behavior of a function refers to the behavior of the graph of the function as it approaches positive infinity or negative infinity.
To determine the end behavior, we first look at the degree of the polynomial function and the sign of the leading coefficient.
The given function is f(x) = 10x^5 + 36x^3 - 42x^2 - 12.
The degree of this polynomial is the highest power of x, which is 5. We then consider the leading coefficient, which is the coefficient of the highest degree term. In this polynomial, the leading coefficient is 10, which is positive.
The end behavior of a function depends on the degree and the sign of the leading coefficient:
1. If the degree is odd and the leading coefficient is positive, the function falls to the left and rises to the right.
2. If the degree is odd and the leading coefficient is negative, the function rises to the left and falls to the right.
3. If the degree is even and the leading coefficient is positive, the function rises to both the left and the right.
4. If the degree is even and the leading coefficient is negative, the function falls to both left and the right.
In our case, the degree is odd (5) and the leading coefficient is positive (10), which means that the function falls to the left and rises to the right.
Therefore, the left end behavior of the function as x approaches negative infinity is negative infinity.