Problem 22 - Precalculus : Building Concepts and Connections

Question

In Exercises \(17-40,\) let \(f(x)=-x^{2}+x, g(x)=\frac{2}{x+1},\) and \(h(x)=-2 x+1 .\) Evaluate each of the following. $$(f+h)(0)$$

Step-by-Step Solution

Answer

The output for \( (f+h)(0) \) is 1.

Step 1: Evaluate f(0)

Evaluate the f function at 0. To do this, replace every instance of x in f with 0, which gives: \(f(0) = -(0)^{2}+(0) = 0\)

Step 2: Evaluate h(0)

Next, evaluate the h function at 0. Replace x in h with 0, to give: \(h(0) = -2*(0)+1 = 1\)

Step 3: Sum the Results

Finally, to solve for (f+h)(0), add together the results of f(0) and h(0): (f+h)(0) = f(0) + h(0) = 0 + 1 = 1

Key Concepts

Function Evaluation

Function evaluation is about finding the value of a function for a given input value. In the exercise, we have functions like \(f(x) = -x^2 + x\) and \(h(x) = -2x + 1\). To evaluate these functions at a specific point, such as 0, you simply plug the number into the function.

• For \(f(0)\), substitute \(x = 0\) to get \(f(0) = -(0)^2 + 0 = 0\).
• For \(h(0)\), substitute \(x = 0\) to get \(h(0) = -2(0) + 1 = 1\).

Function evaluation is straightforward as it involves basic arithmetic manipulation by substituting a value and performing operations as defined by the function. This skill is crucial in mathematics as it forms the basis for understanding more complex operations like composition and solving equations.

Polynomial Functions

Polynomial functions are expressions that involve terms made up of variables raised to whole number exponents and coefficients. These functions take the general form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants. In the exercise, \(f(x) = -x^2 + x\) is a polynomial function.

• Degree: The degree of \(f(x)\) is 2 because the highest power of \(x\) is \(x^2\).
• Leading Coefficient: The leading coefficient, which is the coefficient of the term with the highest power, is \(-1\).
• Constant Term: The constant term is 0, as there is no standalone term without \(x\).

Polynomials can model various phenomena and are central in calculus, algebra, and various applications. They are smooth continuous functions, making them easier to analyze regarding calculus operations like differentiation and integration.

Linear Functions

Linear functions are a type of polynomial function where the highest power of the variable is one. They are represented in the form \(ax + b\), where \(a\) is the slope and \(b\) is the y-intercept. The function \(h(x) = -2x + 1\) from the exercise is a classic linear function.

• Slope: The slope of \(h(x)\) is \(-2\). It tells how steep the line is and in which direction it tilts.
• Y-Intercept: The y-intercept is \(1\). This is the point where the line crosses the y-axis.
• Graph: The graph of a linear function is a straight line.

Linear functions are ubiquitous because they represent constant rates of change and can be found across various fields, such as physics and economics. Their simplicity in expression and analysis makes them one of the foundational concepts in mathematics.