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## End Behaviour of Functions: An Overview

The **end behaviour** of functions describes how a function behaves as the **input values** approach positive or negative infinity. This understanding is crucial for graphing and analysing functions.

### End Behaviour Definition

The **end behaviour** of a function refers to the behaviour of the graph of the function as the input values become very large (\(x \to \infty\)) or very small (\(x \to -\infty\)).

For a polynomial function, the end behaviour is determined by the **leading term**. You can often predict what the graph of a function will do at the ends by considering the degree and leading coefficient of the polynomial.

- If the degree of the polynomial is
**even**and the leading coefficient is**positive**, the ends of the graph will point**upwards**. - If the degree is
**even**and the leading coefficient is**negative**, the ends will point**downwards**. - If the degree is
**odd**and the leading coefficient is**positive**, the graph falls to the left and rises to the right. - If the degree is
**odd**and the leading coefficient is**negative**, the graph rises to the left and falls to the right.

Consider the function \(f(x) = 2x^3 - x^2 + 3x - 1\). The leading term is \(2x^3\):

- Since the degree is
**odd**(3) and the leading coefficient is**positive**(2), as \(x \to \infty\), \(f(x) \to \infty \). - As \(x \to -\infty\), \(f(x) \to -\infty \).

Remember, for non-polynomial functions, you often need to analyse the function more carefully or use limits to determine the end behaviour precisely.

## End Behaviour of Polynomial Functions

The **end behaviour** of polynomial functions refers to the direction the graph of the function heads as the input values approach positive or negative infinity. Understanding this concept is crucial in graphing and analysing polynomial functions.

### Describe the End Behaviour of Polynomial Functions

For polynomial functions, the end behaviour is largely dependent on the **leading term**. The leading term is the term with the highest power of the variable. By evaluating the degree and the coefficient of the leading term, you can predict the end behaviour of the function. Here are the general rules:

- If the degree of the polynomial is
**even**and the leading coefficient is**positive**, the graph will point**upwards**on both ends. - If the degree is
**even**and the leading coefficient is**negative**, the graph will point**downwards**on both ends. - If the degree is
**odd**and the leading coefficient is**positive**, the graph will fall to the left and rise to the right. - If the degree is
**odd**and the leading coefficient is**negative**, the graph will rise to the left and fall to the right.

Consider the polynomial function \(f(x) = -3x^4 + 5x^3 - 2x + 1\). The leading term is \(-3x^4\):

- Since the degree is
**even**(4) and the leading coefficient is**negative**(-3), as \(x \to \infty\), \(f(x) \to -\infty\). - As \(x \to -\infty\), \(f(x) \to -\infty\).

### Example: End Behaviour of Quadratic Functions

Quadratic functions are polynomial functions where the degree is 2. The general form of a quadratic function is \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and the leading term is \(ax^2\). The end behaviour is determined by the leading coefficient **a**:

- If \(a > 0\), the parabola opens upwards, indicating that as \(x \to \infty\), \(f(x) \to \infty\) and as \(x \to -\infty\), \(f(x) \to \infty\).
- If \(a < 0\), the parabola opens downwards, indicating that as \(x \to \infty\), \(f(x) \to -\infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).

For the quadratic function \(f(x) = 2x^2 - 4x + 1\):

- Since \(a = 2 > 0\), as \(x \to \infty\), \(f(x) \to \infty\).
- As \(x \to -\infty\), \(f(x) \to \infty\).

Always look at the leading term to quickly determine the end behaviour of the polynomial function.

### Example: End Behaviour of Cubic Functions

Cubic functions are polynomial functions where the degree is 3. The general form is \(f(x) = ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants. The leading term is \(ax^3\), and the end behaviour depends on the sign of the leading coefficient **a**:

- If \(a > 0\), as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).
- If \(a < 0\), as \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).

Consider the cubic function \(f(x) = -x^3 + 4x^2 - x + 7\):

- Since \(a = -1 < 0\), as \(x \to \infty\), \(f(x) \to -\infty\).
- As \(x \to -\infty\), \(f(x) \to \infty\).

## End Behaviour of Rational Functions

Understanding the **end behaviour** of rational functions is crucial for graphing and interpreting these functions, particularly as the input values approach positive or negative infinity.

### Describe the End Behaviour of Rational Functions

Rational functions are functions of the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. The end behaviour of a rational function depends primarily on the degrees of the numerator \(P(x)\) and the denominator \(Q(x)\).

Degree of \(P(x)\) | Degree of \(Q(x)\) | End Behaviour |

Less than | - | As \(x \to \pm\infty\), \(f(x) \to 0\) |

Equal to | - | As \(x \to \pm \infty\), \(f(x) \to \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients |

Greater than | - | The behaviour will be similar to that of the polynomial quotient \(\frac{P(x)}{Q(x)}\) without any remainders |

Consider the function \(f(x) = \frac{2x^2 + 3}{x^2 - 1}\). The degrees of the numerator (2) and the denominator (2) are equal:

- The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1.
- So, as \(x \to \pm\infty\), \(f(x) \to \frac{2}{1} = 2\).

When the degree of the numerator exceeds the degree of the denominator by exactly one, the end behaviour will feature an oblique asymptote.

### Example: End Behaviour of Proper Rational Functions

A **proper rational function** is a rational function where the degree of the numerator is less than the degree of the denominator. In these cases, as \(x\) approaches infinity, the function will approach zero.

For example, consider the function \(f(x) = \frac{x}{x^2 + 1}\):

- The degree of the numerator is 1, and the degree of the denominator is 2. Since the numerator’s degree is less than the denominator’s degree, as \(x \to \pm\infty\), \(f(x) \to 0\).

### Horizontal and Oblique Asymptotes in End Behaviour

A rational function’s end behaviour is often characterised by **horizontal** or **oblique asymptotes**, depending on the degrees of the polynomial in the numerator and the polynomial in the denominator.

**Horizontal Asymptotes**: If the degree of the numerator is less than or equal to the degree of the denominator, a horizontal asymptote can be determined.**Oblique Asymptotes**: If the degree of the numerator is exactly one more than the degree of the denominator, the function has an oblique (slant) asymptote, determined through polynomial long division.

For instance, consider the function \(f(x) = \frac{x^2 + 1}{x + 2}\):

- The degree of the numerator (2) is one more than the degree of the denominator (1), indicating the presence of an oblique asymptote.
- Performing the division \(\frac{x^2 + 1}{x + 2}\) yields the oblique asymptote \(y = x - 2\).

## End Behaviour of Exponential and Logarithmic Functions

The **end behaviour** of exponential and logarithmic functions gives insight into how these functions behave as their input values approach positive or negative infinity. This understanding is essential when graphing and analysing these types of functions.

### Describe the End Behaviour of Exponential Functions

Exponential functions are functions of the form \(f(x) = a^x\), where \(a\) is a positive constant. The end behaviour of exponential functions can be described as follows:

- If \(a > 1\), as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to 0\).
- If \(0 < a < 1\), as \(x \to \infty\), \(f(x) \to 0\), and as \(x \to -\infty\), \(f(x) \to \infty\).

Consider the function \(f(x) = 3^x\):

- Since \(a = 3 > 1\), as \(x \to \infty\), \(f(x) \to \infty\).
- As \(x \to -\infty\), \(f(x) \to 0\).

Exponential functions often model real-world phenomena like population growth and radioactive decay. For example, the population growth model \(P(t) = P_0 e^{rt}\) describes how a population grows over time, where \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is time.

### Example: End Behaviour of Exponential Growth and Decay

Exponential growth occurs when the base of the exponential function is greater than one, leading to rapid increases over time. Conversely, exponential decay happens when the base is between zero and one, causing the function to decrease over time.

**Exponential growth example:**\(f(x) = 2^x\) – as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to 0\).**Exponential decay example:**\(f(x) = \left(\frac{1}{2}\right)^x\) – as \(x \to \infty\), \(f(x) \to 0\), and as \(x \to -\infty\), \(f(x) \to \infty\).

Remember that the end behaviour of an exponential function is significantly influenced by its base.

### End Behaviour of Logarithmic Functions Explained

Logarithmic functions are functions of the form \(f(x) = \log_a(x)\), where \(a\) is a positive constant. The end behaviour of logarithmic functions shows how they behave as their input values approach positive infinity or zero:

- If \(a > 1\), as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to 0^+\), \(f(x) \to -\infty\).
- If \(0 < a < 1\), as \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to 0^+\), \(f(x) \to \infty\).

**Logarithmic function**: A function of the form \(f(x) = \log_a(x)\), where \(a\) is a positive constant. It is the inverse function of the exponential function \(a^x\).

Consider the function \(f(x) = \log_2(x)\):

- Since \(a = 2 > 1\), as \(x \to \infty\), \(f(x) \to \infty\).
- As \(x \to 0^+\), \(f(x) \to -\infty\).

Logarithmic functions are often used in scientific fields to describe phenomena that scale logarithmically, such as sound intensity (decibels) and the pH level of solutions. They provide a linear way to represent data that cover a wide range of values.

## End behavior of functions - Key takeaways

**End behaviour**: Describes how a function behaves as the input values approach positive or negative infinity.- The end behaviour of
**polynomial functions**is determined by the leading term. Degree and leading coefficient define how the graph's ends point. - The end behaviour of
**rational functions**depends on the degrees of the numerator and denominator, potentially leading to horizontal or oblique asymptotes. - The end behaviour of
**exponential functions**is influenced by the base. For example, if the base is greater than 1, as x approaches infinity, the function approaches infinity. - The end behaviour of
**logarithmic functions**depends on the base. For example, if the base is greater than 1, as x approaches infinity, the function approaches infinity.

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