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Chapter 1: Problem 31

Evaluating a Function In Exercises \(21-32\) , evaluate (if possible) the function at each specified value of the independent variable and simplify. $$ f(x)=\left\\{\begin{array}{ll}{2 x+1,} & {x<0} \\ {2 x+2,} & {x \geq 0}\end{array}\right. $$ $$ (a)f(-1) \quad \text { (b) } f(0) \quad \text { (c) } f(2) $$

Short Answer

Expert verified
The solutions are \(f(-1) = -1\), \(f(0) = 2\), and \(f(2) = 6\).

Step by step solution

01

Evaluate f(-1)

Since -1 is less than 0, use the rule \(2x + 1\). Substituting -1 for \(x\), gives \(f(-1) = 2(-1) + 1 = -2 + 1 = -1)
02

Evaluate f(0)

Since 0 is equal to 0, use the rule \(2x + 2\). Substituting 0 for \(x\), gives \(f(0) = 2(0) + 2 = 0 + 2 = 2)
03

Evaluate f(2)

Since 2 is greater than 0, use the rule \(2x + 2\). Substituting 2 for \(x\), gives \(f(2) = 2(2) + 2 = 4 + 2 = 6)

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Key Concepts

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

Piecewise Functions
In precalculus, a piecewise function is a type of function defined by multiple sub-functions, each of which applies to a specific interval of the function's domain. A piecewise function can be visualized as a mathematical patchwork where each 'piece' of the function has its own rule for calculating outputs based on the inputs within its interval.

Consider the provided exercise, where the function f(x) is defined by two linear equations: one for inputs less than zero (2x + 1) and another for inputs that are zero or greater (2x + 2). This separation into different rules based on the input is what characterizes piecewise functions. They are a powerful tool in mathematics as they can model various situations that change behavior at certain points. Such functions are vital for describing real-world scenarios that are not uniform across an entire range, such as tax brackets, shipping rates, or material strength at different temperatures.
Function Evaluation
The process of function evaluation involves finding the output of a function for a particular input. The input is also known as the independent variable, and the output is the dependent variable. When evaluating functions, you substitute the input value into the function's rule to calculate the corresponding output.

In our example, to evaluate the piecewise function f(x) at x = -1, 0, and 2, we first determine which part of the function to use based on the value of x. After selecting the correct 'piece' of the function, we insert the given input value into its rule for the designated interval. This straightforward procedure allows us to calculate the output or the value of the function at specific points. Understanding how to properly evaluate functions is fundamental in precalculus and carries over into more advanced mathematical concepts.
Precalculus
Precalculus serves as the bridge between algebraic concepts and the more complex world of calculus. It lays the foundation for students to tackle the rigors of calculus, focusing on various functions, their properties, and how to manipulate them algebraically and graphically.

Concepts like piecewise functions and function evaluation are instrumental in precalculus. They allow students to understand how different types of functions behave and how they can be analyzed. Precalculus encourages a deeper comprehension of functions beyond memorizing formulas - it requires students to think critically about how different pieces of a function fit together and affect the overall structure. This big-picture approach to studying functions is crucial as it equips students with the analytical skills needed to solve more complex problems in calculus and beyond.

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Most popular questions from this chapter

Geometry The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width \(x\) surrounds the garden. (a) Draw a diagram that gives a visual representation of the problem. (b) Write the equation for the perimeter \(y\) of the walkway in terms of \(x\) . (c) Use a graphing utility to graph the equation for the perimeter. (d) Determine the slope of the graph in part (c). For each additional one-meter increase in the width of the walkway, determine the increase in its perimeter.

Intercept Form of the Equation of a line, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts \((a, 0)\) and \((0, b)\) is $$\frac{x}{a}+\frac{y}{b}=1, a \neq 0, b \neq 0$$ $$\begin{array}{l}{x \text { -intercept: }\left(\frac{2}{3}, 0\right)} \\ {y \text { -intercept: }(0,-2)}\end{array} $$

Graphical Reasoning Graph each of the functions with a graphing utility. Determine whether the function is even, odd, or neither. $$\begin{array}{ll}{f(x)=x^{2}-x^{4}} & {g(x)=2 x^{3}+1} \\ {h(x)=x^{5}-2 x^{3}+x} & {j(x)=2-x^{6}-x^{8}} \\ {k(x)=x^{5}-2 x^{4}+x-2} & {p(x)=x^{9}+3 x^{5}-x^{3}+x}\end{array}$$ What do you notice about the equations of functions that are odd? What do you notice about the equations of functions that are even? Can you describe a way to identify a function as odd or even by inspecting the equation? Can you describe a way to identify a function as neither odd nor even by inspecting the equation?

Monthly Salary A pharmaceutical salesperson receives a monthly salary of \(\$ 2500\) plus a commission of 7\(\%\) of sales. Write a linear equation for the salesperson's monthly wage \(W\) in terms of monthly sales \(S\) .

Parallel and Perpendicular Lines, determine whether the lines are parallel, perpendicular, or neither. $$\begin{array}{l}{L_{1} : y=\frac{1}{3} x-2} \\ {L_{2} : y=\frac{1}{3} x+3}\end{array}$$
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