Question

Graph the functions $f(x)=x^3,g(x)=3^x$ End $h(x)=x^x$ and find their common intersection point (exactly).

Short answer

Answer: The common intersection point of the three functions is (1,1).

Step-by-step solution

Graphing f(x)

To graph f(x) = x^3, we can plot several points for x and then connect the points. The general shape of the graph of this function is a cubic function that passes through the origin. It is symmetric about the y-axis, and imcreasing for both negative and positive values of x.

Graphing g(x)

To graph g(x) = 3^x, we can plot several points, as well. This is an exponential function that has a horizontal asymptote at y = 0. It is always positive, and it increases at an accelerating rate as x increases.

Graphing h(x)

To graph h(x) = x^x, we can use a graphing calculator or online graphing tool. The function is undefined when x is negative and is valid for non-negative x values. When x = 0, h(x) =1^(1)=1. As x becomes larger, the function increases rapidly.

Overlaying the graphs and finding the intersection points

Now that we have graphed all three functions, we can overlay the graphs to observe their intersection points. We can see that there is one common intersection point for the three functions. To find this point exactly, we need to solve the following equations: 1. f(x) = g(x): $x^3=3^x$ 2. f(x) = h(x) : $x^3=x^x$

Solving the Equations

The first equation, $x^3=3^x$, is a transcendental equation. Solving for x algebraically may be difficult or impossible. Instead, we can use numerical approximation methods, such as the Newton-Raphson method, to approximate the value of x. The second equation, $x^3=x^x$, has $x=1$ as an exact solution since $1^3=1^1$. For any other value of x, the function would either be greater (for values larger than 1) or less (for values smaller than 1) than $x^3$.

Finding the common intersection point

By analyzing the graph of the function and solving the equations, we discovered that there is only one common intersection point of the functions, and it occurs when $x=1$. At this point, the functions have the same value: $f(x)=g(x)=h(x)=1^3=3^1=1^1=1$ Thus, the common intersection point of the three functions is $(1,1)$.

Key concepts

Polynomial Function

Polynomial functions are expressions that involve variables raised to whole number powers. A classic example is the function $f(x)=x^3$. These functions have distinct properties: they are continuous and have a smooth curve with no gaps or jumps. Specifically, the function $f(x)=x^3$ features a cubic polynomial. As this polynomial is of an odd degree, its graph passes through the origin and is symmetric about the origin.

The general shape is like an elongated S curve, stretching from the bottom left to the top right. It's important to note that it increases for both negative and positive values of $x$. To graph this, pick a few values for $x$ (like -2, -1, 0, 1, 2), calculate $f(x)$, and plot these points. Connecting these points will give you the smooth curve of the cubic function.

Exponential Function

Exponential functions are distinguished by a constant base raised to a variable exponent. The function $g(x)=3^x$ illustrates this concept. Unlike polynomial functions, exponential functions show rapid growth or decay. The base number (in this case, 3) determines the rate of increase or decrease. For bases greater than 1, these functions grow rapidly.

The graph of $g(x)=3^x$ has a horizontal asymptote at $y=0$. This means the graph approaches but never touches the x-axis. As $x$ increases, $g(x)$ grows exponentially, curving upward steeply. To visualize this, plot a few points such as $x=-1,0,1,2$ and compute $g(x)$. You'll notice the steep upward curve typical of exponential growth.

Graphing Functions

Graphing functions like $f(x)=x^3$, $g(x)=3^x$, and $h(x)=x^x$ involves finding specific points and trends. Begin with simple points and use them to understand the general behavior of the graph. For $f(x)$ and $g(x)$, plot calculated points and connect them to reveal trends—like the symmetry and rapid growth already mentioned.

For complex functions like $h(x)=x^x$, we may require tools such as graphing calculators or software. $h(x)$ is a fascinating function as it becomes undefined for negative $x$. For other values, like $x=0$, it reduces to $1$. As $x$ increases, the function rises swiftly both in height and complexity, creating a diverse landscape on the graph.

Graphing multiple functions together allows us to visualize their intersections. By overlaying $f(x)$, $g(x)$, and $h(x)$ on the same graph, we spot potential intersection points.

Transcendental Equation

A transcendental equation is one where the variable appears in a non-algebraic manner, such as in exponentials versus polynomials. For instance, solving $x^3=3^x$ involves a cubic polynomial equated with an exponential function. Solving these analytically can be quite complex since traditional algebraic methods often fall short.

One way to tackle them is by using numerical methods like the Newton-Raphson method. These are iterative procedures that approximate solutions by converging ever closer to the true root. Sometimes, such equations are prime candidates for visualization rather than direct computation, using graph intersections to aid in discovering solutions.

The given problem also discusses solving $x^3=x^x$. Here, the equation simplifies at a certain test point, like $x=1$, yielding a solution of $x=1$, because both sides evaluate to 1. This analysis shows us a precise way to deduce solutions particularly for specific values where the equation balances.