Problem 18 - 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. $$(f+g)(0)$$

Step-by-Step Solution

Answer

Therefore, \( (f+g)(0) = 2 \)

Step 1: Understanding the functions

The problem defines two functions \( f(x) = -x^2 + x \) and \( g(x) = \frac{2}{x+1} \). The operation asked is \( f+g \). This operation tells us that we need to add \( f(x) \) and \( g(x) \) together.

Step 2: Combine the functions

When we have the function \( (f+g)(x) \), it means \( f(x) + g(x) \). So, it becomes \( (-x^{2}+x) + \frac{2}{x+1} \) which simplifies to \( -x^{2} +x + \frac{2}{x+1} \)

Step 3: Substitute x with 0

Now, we need to substitute 0 into our x values. That gives \( -(0)^{2} + 0 + \frac{2}{0+1} \) which simplifies to \( 0 + 0 + 2 \)

Key Concepts

Combining Functions

When working with multiple functions in precalculus, a fundamental skill is to combine them into a single resulting function. This process is similar to combining like terms in an algebraic expression but involves applying operations to entire functions. To combine functions, we apply an operation, such as addition, subtraction, multiplication, or division, between two functions.

For instance, if you are given two functions, say, \( f(x) = -x^2 + x \) and \( g(x) = \frac{2}{x+1} \), and you need to find \( (f+g)(x) \), you simply add the expressions for \( f(x) \) and \( g(x) \) together. It is crucial to apply any necessary algebraic simplifications after combining them, such as combining like terms or simplifying fractions, to obtain the most direct form of the new function.

Function Evaluation

Evaluating a function means finding its value at a specific point, which is the essence of function application in mathematics. To evaluate a function, replace every occurrence of the function variable with the given number or expression, and then simplify.

Take for example the combined function we just discussed, \( -x^{2} +x + \frac{2}{x+1} \). To evaluate this at \( x = 0 \), we substitute 0 in place of every \( x \). This results in \( -(0)^{2} + 0 + \frac{2}{0+1} \), which simplifies down to 2. This process is critical for understanding how a function behaves at specific values and is particularly important in applications such as graphing functions or solving real-world problems.

Polynomial Functions

Polynomial functions are algebraic expressions involving a sum of powers of the variable, each multiplied by a coefficient. The general form of a polynomial function is \( a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0 \), where the \( a_i \) are constants, and \( n \) is a non-negative integer that represents the degree of the polynomial.

Functions like \( f(x) = -x^2 + x \) in our example are polynomial functions. They are particularly important due to their straightforward properties and the ease with which they can be manipulated algebraically. Analyzing and graphing polynomial functions provide key insights into their limits, continuity, and differentiability among other concepts, thereby forming a fundamental aspect of precalculus and calculus studies.