Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate 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) \(f(-1)\) (b) \(f(0)\) (c) \(f(2)\)

Step-by-Step Solution

Answer

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

Step 1: Evaluate f(-1)

Since -1 is less than 0, we will use the expression for \(f(x)\) that's defined for \(x<0\), which is \(2x+1\). Substituting -1 into this expression, we obtain: \(f(-1) = 2(-1) + 1 = -2 + 1 = -1 \).

Step 2: Evaluate f(0)

0 is equal to 0, so we use the expression for \(f(x)\) that's defined for \(x \geq 0\), which is \(2x+2\). Substituting 0 into this expression, we obtain: \(f(0) = 2(0) + 2 = 0 + 2 = 2 \).

Step 3: Evaluate f(2)

Since 2 is greater than 0, we use the expression for \(f(x)\) that's defined for \(x \geq 0\), which is \(2x+2\). Substituting 2 into this expression, we obtain: \(f(2) = 2(2) + 2 = 4 + 2 = 6 \).

Key Concepts

Evaluating Functions

When dealing with mathematical problems, one of the common tasks is evaluating functions. This process involves substituting a specific value into a function and performing the necessary calculations to find the result.

For instance, if we have a function like the piecewise function given in the exercise \(f(x)\), the first step is to decide which piece of the function applies to the value of \(x\) we want to evaluate. In the given example, there are two pieces: one for \(x<0\) and another for \(x\geq0\).

After determining the correct piece to use, we then substitute the value of \(x\) into the selected expression and simplify it to find \(f(x)\). For example, to evaluate \(f(-1)\), we use the part of the function for \(x<0\) and substitute \(x\) with -1, resulting in a simplified answer of -1.

Function Notation

Understanding function notation is crucial in algebra and higher-level mathematics. Function notation, using symbols like \(f(x)\), is a concise way to represent the outputs of a function corresponding to its inputs. It tells us that \(f\) is the name of the function and \(x\) is the variable.

In piecewise functions, which are just functions defined by different expressions depending on the value of \(x\), function notation helps specify which part of the function to use. As seen in the exercise, the function \(f\) depends on whether \(x\) is less than or equal to, or greater than zero. This is reflected in how the function is written using a curly brace to define the different conditions.

For example, the notation \(f(0)\) tells us to evaluate the function \(f\) when \(x\) is 0. Here, we look at the function definition and see that the condition for \(x\geq0\) must be used, leading us to calculate the result as 2.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They do not contain an equal sign, as equations do, and are used to represent values in functions.

In the context of evaluating functions and piecewise functions, algebraic expressions become pieces of a larger puzzle where each piece corresponds to a particular condition or interval. If we consider the given exercise, \(2x+1\) and \(2x+2\) are two different algebraic expressions that make up the function \(f(x)\).

These expressions can be simplified when a specific value is substituted for the variable \(x\). It's like following a recipe: the expression gives the ingredients and the instructions, and the value you plug in is like adding a specific ingredient. The art of simplifying these expressions is at the heart of algebra and is fundamental for problem-solving in mathematics.