Consider, j(x) = 12x^3/5 - 27x/5. Describe and provide an explanation for the end behavior of j(x).
To determine the end behavior of the function j(x) = 12x^(3/5) - 27x/5, we can analyze the leading term as x approaches positive and negative infinity.
The leading term in this case is 12x^(3/5) as it has the highest power of x.
1. As x approaches positive infinity:
As x becomes larger and larger positive values, the leading term 12x^(3/5) dominates the equation. Since x^(3/5) will always be positive regardless of the value of x, the product of 12 and x^(3/5) will also be positive. Hence, as x approaches positive infinity, j(x) will tend towards positive infinity.
2. As x approaches negative infinity:
As x becomes more and more negative, the leading term 12x^(3/5) will also become negative, as raising a negative number to an odd fractional power results in a negative value. Furthermore, the coefficient of the second term is negative (-27/5), so the second term will also be negative. Since both terms contribute to the negative values, as x approaches negative infinity, j(x) will tend towards negative infinity.
In conclusion, as x approaches positive infinity, j(x) will tend towards positive infinity, and as x approaches negative infinity, j(x) will tend towards negative infinity.
To describe the end behavior of a function, we look at the behavior of the function as x approaches positive infinity and negative infinity.
Let's analyze the function j(x) = 12x^3/5 - 27x/5.
As x approaches positive infinity (+∞), we need to determine the behavior of j(x). We are interested in the highest power term in the function because it will dominate the behavior for large values of x. In this case, the highest power term is 12x^3/5.
Since the exponent of x is 3/5 (which is greater than 0 but less than 1), we know that the function will approach positive infinity as x approaches positive infinity. This means that as x becomes larger and larger, j(x) will also become larger and larger.
Now let's consider the behavior of j(x) as x approaches negative infinity (-∞). Again, we focus on the highest power term which is 12x^3/5.
For negative values of x, we need to look at the exponent 3/5. Since 3/5 is an odd power, the function will have different behaviors for negative and positive values. Specifically, as x approaches negative infinity, the behavior will be opposite to that of positive infinity.
For example, if we substitute a very large negative number into the function, we will have a negative value, and as we make x more negative, the function will become more negative as well.
To summarize, as x approaches positive infinity, j(x) approaches positive infinity, and as x approaches negative infinity, j(x) approaches negative infinity.
To describe the end behavior of a function, we analyze what happens to the function as x approaches positive infinity and negative infinity.
The given function is j(x) = 12x^(3/5) - 27x/5.
As x approaches positive infinity, the dominant term in the function is 12x^(3/5). Since the exponent is less than 1, this term will approach zero as x gets larger. The term -27x/5 will also approach zero because the coefficient (-27/5) is multiplied by x.
Therefore, as x approaches positive infinity, j(x) approaches zero. This means that the graph of j(x) will approach the x-axis from above.
Similarly, as x approaches negative infinity, both the terms in the function will approach zero. Therefore, j(x) also approaches zero as x approaches negative infinity. This indicates that the graph of j(x) will approach the x-axis from below.
In conclusion, the end behavior of j(x) is that the graph approaches the x-axis both from above as x approaches positive infinity and from below as x approaches negative infinity.