Functions And Intersections Revision Continued

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Functions are a fundamental concept in mathematics, serving as the building blocks for modeling real-world phenomena and solving complex problems. In this comprehensive revision, we delve deeper into the intricacies of functions, focusing on their algebraic representations, graphical interpretations, and intersections. Understanding how different functions interact with each other is crucial for advanced mathematical studies and practical applications in various fields. This article aims to provide a thorough review of key concepts related to functions, equipping you with the skills to tackle challenging problems and enhance your mathematical proficiency.

6. Analyzing Functions and Finding Points of Intersection

a) Algebraic Determination of Intersection Points

To find the coordinates of the point(s) of intersection between two functions algebraically, we need to solve the system of equations formed by setting the function expressions equal to each other. In this case, we are given three functions:

  • f(x)=12(x+3)(x−3)f(x) = \frac{1}{2}(x+3)(x-3)

  • g(x)=−4xg(x) = -\frac{4}{x}

  • h(x)=−2x+2h(x) = -2x + 2

We are tasked with finding the intersection points of $g(x)$ and $h(x)$. To do this, we set $g(x)$ equal to $h(x)$ and solve for $x$:

−4x=−2x+2- \frac{4}{x} = -2x + 2

To eliminate the fraction, we multiply both sides of the equation by $x$:

−4=−2x2+2x-4 = -2x^2 + 2x

Now, rearrange the equation into a quadratic form:

2x2−2x−4=02x^2 - 2x - 4 = 0

Divide the entire equation by 2 to simplify:

x2−x−2=0x^2 - x - 2 = 0

This quadratic equation can be factored as follows:

(x−2)(x+1)=0(x - 2)(x + 1) = 0

Setting each factor to zero gives us the solutions for $x$:

x−2=0⇒x=2x - 2 = 0 \Rightarrow x = 2

x+1=0⇒x=−1x + 1 = 0 \Rightarrow x = -1

Now that we have the $x$-coordinates of the intersection points, we can find the corresponding $y$-coordinates by substituting these values back into either $g(x)$ or $h(x)$. Let's use $h(x) = -2x + 2$:

For $x = 2$:

h(2)=−2(2)+2=−4+2=−2h(2) = -2(2) + 2 = -4 + 2 = -2

So, one point of intersection is $(2, -2)$.

For $x = -1$:

h(−1)=−2(−1)+2=2+2=4h(-1) = -2(-1) + 2 = 2 + 2 = 4

Thus, the other point of intersection is $(-1, 4)$.

Therefore, the coordinates of the points of intersection of $g(x)$ and $h(x)$ are $(2, -2)$ and $(-1, 4)$. This algebraic method allows us to precisely determine the points where two functions meet, providing valuable insights into their relationships.

b) Sketching the Graphs and Verifying Intersections

Sketching the graphs of the functions $g(x)$ and $h(x)$ provides a visual representation of their intersection points, allowing us to verify the algebraic solutions and gain a deeper understanding of their behavior. To sketch these graphs, we need to understand the basic shapes and characteristics of each function.

Sketching $g(x) = -\frac{4}{x}$

The function $g(x) = -\frac{4}{x}$ is a rational function, which is a function that can be written as the ratio of two polynomials. In this case, the numerator is -4 and the denominator is $x$. Rational functions have several key features:

  • Vertical Asymptote: A vertical asymptote occurs where the denominator of the rational function is zero. For $g(x)$, the denominator is $x$, so there is a vertical asymptote at $x = 0$. This means the function approaches infinity (or negative infinity) as $x$ approaches 0.
  • Horizontal Asymptote: A horizontal asymptote describes the behavior of the function as $x$ approaches positive or negative infinity. For $g(x)$, as $|x|$ becomes very large, the value of $-\frac{4}{x}$ approaches 0. Therefore, there is a horizontal asymptote at $y = 0$.
  • Shape of the Graph: The graph of $g(x)$ is a hyperbola. Since the numerator is negative, the graph will be in the second and fourth quadrants. This means that for positive $x$, the function values are negative, and for negative $x$, the function values are positive.

To sketch the graph, we can plot a few key points. For example:

  • When $x = 1$, $g(1) = -4$
  • When $x = 2$, $g(2) = -2$
  • When $x = 4$, $g(4) = -1$
  • When $x = -1$, $g(-1) = 4$
  • When $x = -2$, $g(-2) = 2$
  • When $x = -4$, $g(-4) = 1$

Plotting these points and considering the asymptotes, we can sketch the graph of $g(x)$, which is a hyperbola in the second and fourth quadrants.

Sketching $h(x) = -2x + 2$

The function $h(x) = -2x + 2$ is a linear function, which means its graph is a straight line. Linear functions are easy to sketch because they are defined by their slope and y-intercept:

  • Slope: The slope of the line is the coefficient of $x$, which is -2. A negative slope means the line decreases as $x$ increases.
  • Y-intercept: The y-intercept is the value of $y$ when $x = 0$. In this case, the y-intercept is 2.

To sketch the graph, we can plot the y-intercept at $(0, 2)$ and use the slope to find another point. Since the slope is -2, for every 1 unit increase in $x$, the value of $y$ decreases by 2. So, if we start at $(0, 2)$ and move 1 unit to the right, we move 2 units down to the point $(1, 0)$.

Connecting these two points gives us the line representing $h(x)$.

Verifying the Intersection Points

By sketching both graphs on the same coordinate plane, we can visually verify the intersection points we found algebraically. The graph of $g(x)$ (the hyperbola) and the graph of $h(x)$ (the line) will intersect at two points. These points should correspond to the coordinates we calculated earlier: $(2, -2)$ and $(-1, 4)$.

The graphical representation provides a clear confirmation of our algebraic solutions. If the graphs intersect at the calculated points, it validates our calculations and deepens our understanding of the functions' behavior.

In conclusion, sketching the graphs of the functions is an essential step in analyzing their relationships and verifying algebraic solutions. It offers a visual context that enhances our comprehension of function behavior and intersection points.

This detailed explanation and analysis of functions and their intersections should provide a comprehensive understanding of the topic, making it easier to grasp the concepts and apply them effectively. By combining algebraic methods with graphical representations, we can gain a complete and accurate picture of function behavior and their interactions. This dual approach is crucial for mastering mathematical concepts and tackling complex problems.