Problem 12 - Calculus Volume 1

Question

For the following exercises, find the values for each function, if they exist, then simplify. a. \(f(0)\) b. \(f(1)\) c. \(f(3)\) d. \(f(-x)\) e. \(f(a)\) f. \(f(a+h)\) $$f(x)=\frac{x-2}{3 x+7}$$

Step-by-Step Solution

Answer

a) \(-\frac{2}{7}\); b) \(-\frac{1}{10}\); c) \(\frac{1}{16}\); d) \(\frac{-x-2}{-3x+7}\); e) \(\frac{a-2}{3a+7}\); f) \(\frac{a+h-2}{3a+3h+7}\).

Step 1: Evaluate f(0)

To find \(f(0)\), substitute \(x = 0\) in the function \(f(x) = \frac{x-2}{3x+7}\):\[f(0) = \frac{0-2}{3(0)+7} = \frac{-2}{7}\]The value of \(f(0)\) is \(-\frac{2}{7}\).

Step 2: Evaluate f(1)

To find \(f(1)\), substitute \(x = 1\) in the function:\[f(1) = \frac{1-2}{3(1)+7} = \frac{-1}{10}\]The value of \(f(1)\) is \(-\frac{1}{10}\).

Step 3: Evaluate f(3)

To find \(f(3)\), substitute \(x = 3\) in the function:\[f(3) = \frac{3-2}{3(3)+7} = \frac{1}{16}\]The value of \(f(3)\) is \(\frac{1}{16}\).

Step 4: Evaluate f(-x)

To find \(f(-x)\), substitute \(x = -x\) in the function:\[f(-x) = \frac{-x-2}{3(-x)+7} = \frac{-x-2}{-3x+7}\]The expression for \(f(-x)\) is \(\frac{-x-2}{-3x+7}\).

Step 5: Evaluate f(a)

To find \(f(a)\), substitute \(x = a\) in the function:\[f(a) = \frac{a-2}{3a+7}\]The expression for \(f(a)\) is \(\frac{a-2}{3a+7}\).

Step 6: Evaluate f(a+h)

To find \(f(a+h)\), substitute \(x = a+h\) in the function:\[f(a+h) = \frac{(a+h)-2}{3(a+h)+7} = \frac{a+h-2}{3a + 3h + 7}\]The expression for \(f(a+h)\) is \(\frac{a+h-2}{3a + 3h + 7}\).

Key Concepts

Substitution in Functions

Substitution in functions is a fundamental concept in mathematics where you replace the variable in a function with a given value. This allows us to evaluate the function at that specific point. In the original exercise, the function provided was a rational function:

\[ f(x) = \frac{x-2}{3x+7} \]To find values like \(f(0)\) or \(f(1)\), you're asked to substitute these numbers directly into the place of \(x\). For example:- \(f(0) = \frac{0-2}{3(0)+7} = \frac{-2}{7}\).- \(f(1) = \frac{1-2}{3(1)+7} = \frac{-1}{10}\).This method can also be applied to variables, such as substituting \(x\) with \(-x\), \(a\), or \(a+h\) to explore how the function behaves with different inputs.

Simplification of Expressions

Simplification of expressions is the process of reducing a mathematical expression to its simplest form. This often involves arithmetic operations and algebraic transformations.After substituting a value into a function, the expression that results may sometimes need simplification. For instance, when evaluating \(f(0)\) from the original function:\[ f(0) = \frac{0-2}{3(0)+7} = \frac{-2}{7} \] We see that subtraction and addition are used to simplify the expression. Simplification not only makes expressions easier to read and understand, but it also allows for more straightforward comparisons between expressions. This can involve combining like terms, factoring, and using arithmetic to reduce a rational expression, which is often seen in evaluating functions like \(f(x)\).

Rational Functions

Rational functions are a type of function represented by the quotient of two polynomials. A central feature of rational functions is that they can have undefined points, where the denominator equals zero.The function \(f(x) = \frac{x-2}{3x+7}\) is a rational function. Understanding this type of function implies knowing how to handle potential restrictions, as the function is undefined if the denominator is zero. Fortunately, in this exercise, there are no such values for \(x\) that make \(3x + 7 = 0\) since solving for \(x\) results in a non-real number.Rational functions are important for modeling real-world situations where rates, ratios, or proportions play a role. They are often used in calculus for limits, where analyzing behaviors as functions approach certain x-values is crucial.

Function Notation

Function notation is a way of expressing a function in order to show its input-output relationship clearly. It typically involves symbols like \(f(x)\), where "\(f\)" is the name of the function, and "\(x\)" represents the variable input.In the exercise, the notation \(f(x) = \frac{x-2}{3x+7}\) describes a function \(f\) with an input \(x\). When you see \(f(0)\), this indicates that the input is \(0\), and you substitute this value into the function. Function notation provides an efficient way to communicate complex concepts with a simple structure. Understanding it allows one to easily interpret and manipulate functions across various branches of math, including algebra and calculus.Having a good grasp of function notation is essential, as it's used to define, evaluate, and communicate mathematical functions in a uniform manner.