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Chapter 5: Problem 9

Sketch a complete graph of the function. $$h(x)=2^{x^{2}}$$

Short Answer

Expert verified
Question: State the domain, range, intercepts, and symmetry of the function $$h(x) = 2^{x^2}$$ and describe its end behavior. Answer: The function $$h(x) = 2^{x^2}$$ has a domain of $$(-\infty, \infty)$$ and a range of $$(0, \infty)$$. It has no x-intercepts and has a y-intercept at the point $$(0,1)$$. It is an even function, which means it is symmetric about the y-axis. As x approaches $$\pm\infty$$, the function grows without bound, showing that there are no asymptotes and the graph increases steeply for positive x values and converges to the y-axis as x approaches negative infinity.

Step by step solution

01

Determine the domain and range

Since the base of the exponential function is $$2 > 1$$, and the exponent includes a square, the function will be always positive. This means that the graph will be above the x-axis for all values of x. - Domain: The function is defined for all values of x, so the domain is $$(-\infty, \infty)$$. - Range: Since the function is always positive, the range is $$(0, \infty)$$.
02

Find the intercepts

- For x-intercept: When $$y = 0$$, we have $$0 = 2^{x^2}$$. But since the function is always positive, there is no x-intercept. - For y-intercept: When $$x = 0$$, we have $$h(0) = 2^{0^2} = 2^0 = 1$$. So, the y-intercept is at the point $$(0,1)$$.
03

Determine the symmetry

We will check if the function is an even, odd, or neither function by analyzing $$h(-x)$$. $$h(-x) = 2^{(-x)^2} = 2^{x^2} = h(x)$$ Since $$h(-x) = h(x)$$, the function is an even function, and the graph is symmetric about the y-axis.
04

Find asymptotes and end behavior

- Horizontal asymptote: As x approaches $$\pm\infty$$, the exponent $$x^2$$ will always be positive and increasing, causing the function value to approach infinity. So, there is no horizontal asymptote. - Vertical asymptote: Since the function is defined for all values of x, there is no vertical asymptote.
05

Sketch the graph

Using the above information, we can sketch the graph of $$h(x) = 2^{x^2}$$: 1. Plot the y-intercept at $$(0,1)$$. 2. Since the graph is symmetric about the y-axis and the range is $$(0,\infty)$$, the graph will always be above the x-axis and symmetric with respect to the y-axis. 3. Sketch the graph by showing it is increasing steeply for positive x values, but converging to the y-axis as x approaches negative infinity. 4. As x approaches $$\pm\infty$$, the graph grows without bound, and there are no asymptotes. We now have a complete sketch of the function $$h(x) = 2^{x^2}$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range of a Function
Understanding the domain and range of a function is fundamental in graphing. The domain refers to all possible input values (x-values) for which the function is defined. For exponential functions, such as \( h(x) = 2^{x^2} \), the domain is typically all real numbers, denoted by \( (-fty, fty) \), since the exponent can be any real number.

The range, on the other hand, represents all possible output values (y-values) the function can produce. In our example, because an exponential function with a positive base greater than one grows without bound and never becomes negative or zero, the range is \( (0, fty) \). Thus, the function's graph will always be above the x-axis, reflecting that positive output.
Exponential Function Properties
Exponential functions, such as \( h(x) = 2^{x^2} \), exhibit distinct properties. Primarily, they grow at a rate proportional to their current value, which explains their steep increase. These functions have a constant base, in this case, 2, raised to a variable exponent. Since 2 is greater than 1, the function will increase as x increases.

Growth behavior varies with the exponent: when the exponent \( x^2 \) increases, as x moves away from zero in either direction, the function's value grows exponentially. Moreover, exponential functions never touch the x-axis when the base is greater than one, which means they have no x-intercept, as represented in our example.
Symmetry in Functions
Symmetry in a graph can provide insights into a function's behavior. For the function \( h(x) = 2^{x^2} \), symmetry plays a key role. It is an even function, confirming this with the property \( h(-x) = h(x) \).

Graphically, this means the function's graph is mirrored across the y-axis or the line \( x = 0 \). When graphing, plotting points on one side of the y-axis and reflecting them across can simplify your work. Identifying symmetry helps ensure accuracy and uniformity in your sketch of the function's graph and reduces the potential for errors.
Function Intercepts
Intercepts are where the graph of a function crosses the axes. Identifying intercepts is a step towards building a clear picture of the function's graph. In our equation's case, \( h(x) = 2^{x^2} \), it's clear that the function cannot have an x-intercept because the output is never zero for any real number x.

The y-intercept is found when \( x = 0 \). For \( h(x) \), substituting \( x \) with zero yields \( h(0) = 2^{0^2} = 2^0 = 1 \), resulting in a y-intercept at the point \( (0,1) \). This point is critical as it is one of the few precise locations on the graph we can determine exactly without estimation.

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