Question

Use a graphing calculator to evaluate \((f+g)(x),(f-g)(x),(f g)(x)\), and \(\left(\frac{f}{g}\right)(x)\) when \(x=5\). Round your answer to two decimal places. $$ f(x)=-2 x^{1 / 3} ; g(x)=5 x^{1 / 2} $$

Short answer

(f+g)(5) = 7.76, (f-g)(5) = -14.60, (fg)(5) = -38.20, (f/g)(5) = -0.31.

Step-by-step solution

Find (f+g)(x) at x = 5

(f+g)(5) = f(5) + g(5)\n= -2*5^(1/3) + 5*5^(1/2)\n= -2*1.71 + 5*2.236\n= -3.42 + 11.18\n= 7.76

Find (f-g)(x) at x = 5

(f-g)(5) = f(5) - g(5)\n= -2*5^(1/3) - 5*5^(1/2)\n= -2*1.71 - 5*2.236\n= -3.42 - 11.18\n= -14.60

Find (fg)(x) at x = 5

(fg)(5) = f(5) * g(5)\n= -2*5^(1/3) * 5*5^(1/2)\n= -2*1.71 * 5*2.236\n= -3.42 * 11.18\n= -38.20

Find f/g(x) at x = 5

(f/g)(5) = f(5) / g(5)\n= -2*5^(1/3) / 5*5^(1/2)\n= -2*1.71 / 5*2.236\n= -3.42 / 11.18\n= -0.31

Key concepts

Graphing Calculators

Graphing calculators are a powerful tool for students and mathematicians alike. They allow us to visualize functions and calculate their values, which can be especially helpful for complex operations like function addition or multiplication. Graphing calculators can handle a variety of tasks:

• Plotting multiple functions on a single graph
• Calculating function values at specific points
• Performing operations like addition, subtraction, multiplication, and division on functions

Using a graphing calculator, you can easily evaluate and visualize \( (f+g)(x), (f-g)(x), (f \cdot g)(x), \text{ and } \left(\frac{f}{g}\right)(x) \).Visualizing these can help you understand the behavior and intersections of functions. Graphing calculators are not only efficient but also enhance understanding by offering a clear picture of function dynamics.

Function Addition and Subtraction

Function addition and subtraction are straightforward operations that involve combining the outputs of two functions. Suppose you have two functions, \( f(x) \) and \( g(x) \). The addition of these functions is shown as \((f+g)(x) = f(x) + g(x)\). Similarly, subtraction is \((f-g)(x) = f(x) - g(x)\). Here's how you can work with them:

• Calculate the value of each function separately at a given point.
• Add or subtract the results as needed.

In our exercise, we calculated these at \( x=5 \) and combined them to get the result. Function operations like these are not only useful in algebra, but they're also foundational in calculus and other advanced math applications. By practicing these techniques, you can build a strong foundation for more complex mathematical concepts.

Function Multiplication and Division

Function multiplication and division might seem intimidating at first, but they are just as manageable as addition and subtraction.

• For multiplication, you multiply the output of two functions at a point: \((f \cdot g)(x) = f(x) \cdot g(x)\).
• For division, you divide the output of two functions: \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). Make sure \(g(x)\) is not zero to avoid undefined results.

In this exercise, when calculating at \( x=5 \), the multiplication and division of the functions provided insights into their combined growth or decay trends. These operations are crucial in understanding more complex function behavior, especially in calculus where they are part of derivative and integration processes.

Evaluating Functions

Evaluating functions is fundamental to understanding how they work. It involves substituting a particular value into the function and solving it to get a result. Here is how you do it:

• Identify the function you need to evaluate, such as \( f(x) = -2x^{1/3} \)
• Substitute the given value for \( x \) into the function. For example, if \( x=5 \), replace every \( x \) in \( f(x) \) with 5.
• Solve the expression using arithmetic and possibly a calculator for precision.

By evaluating functions at specific points, you gain insights into their numerical values and how they behave. This practice is not only essential for basic mathematics; it also paves the way for dealing with intricate problems in pure and applied mathematics.