Question

a) Use technology to graph each function and identify the domain and range. \(f(x)=x^{2}+4\) ii) \(g(x)=x^{2}-4\) iii) \(h(x)=-x^{2}+4\) iv) \(j(x)=-x^{2}-4\) b) Graph the square root of each function in part a) using technology. c) What do you notice about the graph of \(y=\sqrt{j(x)} ?\) Explain this observation based on the graph of \(y=j(x) .\) Then, explain this observation algebraically. d) In general, how are the domains of the functions in part a) related to the domains of the functions in part b)? How are the ranges related?

Short answer

Domains in part a) are \(( -\infty, \infty )\), but restricted in part b) to where the functions are non-negative. Graph of \(\sqrt{j(x)}\) does not exist because \(j(x)\) is always negative.

Step-by-step solution

Graph each function

Using graphing technology (e.g., a graphing calculator or computer software), plot the following functions:- For \(f(x) = x^2 + 4\)- For \(g(x) = x^2 - 4\)- For \(h(x) = -x^2 + 4\)- For \(j(x) = -x^2 - 4\)Identify the domain and range for each: - Domain of all these functions is \((-\infty, \infty)\).- Range of \(f(x)\) is \([4, \infty)\).- Range of \(g(x)\) is \([-4, \infty)\).- Range of \(h(x)\) is \(( -\infty, 4 ]\).- Range of \(j(x)\) is \(( -\infty, -4 ]\).

Graph the square root of each function

Using graphing technology, graph the square root of each function from part a) and analyze:- \(y = \sqrt{f(x)}\)- \(y = \sqrt{g(x)}\)- \(y = \sqrt{h(x)}\)- \(y = \sqrt{j(x)}\)Note that the square root function is only defined where the original function is non-negative.

Observe the graph of \(y = \sqrt{j(x)}\)

Graph where \(j(x)\) is non-negative and compare to the original graph:\(y = \sqrt{j(x)}\) will have no values since \(j(x)\) is always negative (\( -x^2 -4 \le -4\)). Thus, the graph of \(\sqrt{j(x)}\) does not exist in the real plane.

Analyze the domain and range relationships

Examine how the domains and ranges have transformed for each pair:Domains in part a) are typically all real numbers \(( -\infty, \infty )\), while the domains in part b) depend on where the original functions are non-negative.The ranges in part a) are intervals, while in part b), the ranges are typically intervals starting from zero.

Key concepts

Domain and Range

When understanding functions, it is crucial to grasp the concepts of domain and range. The domain of a function is the complete set of possible values of the independent variable, usually represented as 'x'. In simpler terms, it is all the values that you can input into the function. For example, for the functions like the ones in our exercise - such as \(f(x) = x^2 + 4\) - the domain is all real numbers since you can plug any number into 'x' and still get a valid output.

On the other hand, the range of a function is the complete set of all possible values of the dependent variable usually represented as 'y'. It's all the values that the function can output. For instance, \(f(x) = x^2 + 4\) has a range starting from 4 and goes to infinity, because \(x^2\) is always non-negative, and adding 4 shifts the minimum value up to 4. In our exercise, the domain for all functions \(f(x), g(x), h(x), \text{and} j(x)\) is \((-\infty, \infty)\), but their ranges differ depending on the function structure.

Understanding these concepts helps to classify and analyze different functions effectively.

Square Root Functions

Square root functions are particular types of functions where the variable is under a square root symbol. These functions have unique characteristics. One crucial aspect is that square root functions are only defined for non-negative values under the square root sign. This is because the square root of a negative number is not defined in the real number system.

In our exercise, we need to graph the square root of each given function from part a). For example, \(y = \sqrt{f(x)}\). Since \(f(x) = x^2 + 4\), we can only take the square root of this function wherever \(x^2 + 4 > 0\). However, since this expression is always positive (as the smallest value it can take is 4), the square root function can be defined for all x values. But for a function like \ j(x) = -x^2 - 4\, it is always negative, and thus, \(y = \sqrt{j(x)}\) has no real values, leading to no graph existing for this function in the real plane.

Graphing Technology

Graphing technology such as graphing calculators or computer software plays a significant role in visualizing functions. These tools help plot complicated functions efficiently and accurately, making it easier to interpret results and analyze different aspects such as domain and range.

Using graphing tools, we can plot the equations from our exercise with ease. For instance, plotting \(f(x) = x^2 + 4\), you'll see a parabola that opens upwards, starting from the point (0,4). Similarly, \ g(x) = x^2 - 4 \ starts at point (0,-4). Using these tools to graph square root functions like \(y = \sqrt{g(x)}\) helps to clearly understand how the graph behaves, showing a parabola starting from zero but not extending to negative y-values due to the restriction of square root functions. Hence, technology enhances the learning and understanding process by providing a graphical representation of mathematical concepts.

Non-negative Functions

Non-negative functions are those that never take on negative values. This property is crucial when dealing with square root functions since the square root of a negative number is not defined in real numbers.

For instance, in our exercise, when we deal with \(h(x) = -x^2 + 4\), we know the expression \ -x^2 + 4 \ is non-negative only when \ x^2 \is less than or equal to 4, or equivalently for \ |x| \ less than or equal to 2. When we attempt to graph \(y = \sqrt{h(x)}\), the function will only exist between \-2 \ and \ 2 \, thus emphasizing the non-negative part of the function.

Therefore, understanding and being able to identify non-negative regions of functions is essential for correctly graphing square root functions and comprehending their domain and range relationships. This understanding directly relates to the observation that \ y = \sqrt{j(x)} \ does not exist in the real plane because the function \ j(x) = -x^2 - 4 \ is always negative.