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Chapter 1: Problem 20

Evaluate the indicated function, where \(f(x)=x^{2}-3 x+2\) and \(g(x)=2 x-4\). $$(f-g)(0)$$

Short Answer

Expert verified
The answer is 6

Step by step solution

01

Evaluate \(f(0)\)

Plugging in 0 into \(f(x)\), we get \(f(0) = (0)^2 - 3(0) + 2 = 2\).
02

Evaluate \(g(0)\)

Similarly, we plug in 0 into \(g(x)\), giving us \(g(0) = 2(0) - 4 = -4\).
03

Evaluate \((f-g)(0)\)

Now, we figure out \((f-g)(0)\) by subtracting the result from \(g(0)\) from that of \(f(0)\), which is \(2 - (-4) = 6\).

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Key Concepts

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

Function Evaluation
Understanding how to evaluate a function is a fundamental skill in precalculus. Function evaluation is the process of determining the output of a function given a specific input. It requires substituting the input value into the function and performing the operations within.

In our exercise, function evaluation is demonstrated when we replace the variable x with the value 0 in the functions given by:
  • For the function f(x) = x^2 - 3x + 2, the evaluation at x = 0 yields f(0) = 0^2 - 3(0) + 2, which simplifies to f(0) = 2.
  • Similarly, for g(x) = 2x - 4, we find g(0) = 2(0) - 4, simplifying to g(0) = -4.
The procedure follows a simple replace-and-simplify methodology. It is important not to skip any steps and to ensure all operations are performed according to the order of operations. This step-by-step approach prevents mistakes and solidifies understanding of the function's behavior.
Difference of Functions
In precalculus, learning about operations on functions, including the difference of functions, is crucial. The difference of two functions, commonly expressed as (f - g)(x), involves subtracting the value of one function from another function for the same input x.

To calculate the difference:
  • First evaluate each function separately at the specific value of x, as we did in the previous concept.
  • Then simply subtract the result of g(x) from f(x).
In our exercise, the difference at x = 0 (denoted as (f - g)(0)) involves the subtraction of g(0) from f(0), which is 2 - (-4) and results in 6. The operation may seem straightforward, but precision is essential to avoid arithmetic errors, especially when dealing with negative numbers.
Polynomial Functions
Polynomial functions form a significant concept in precalculus. Defined by terms consisting of variables raised to whole number exponents and their coefficients, these functions are represented in the form of a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0, where each a represents a coefficient and n is a non-negative integer.

In this exercise, both given functions are examples of polynomial functions. The function f(x) = x^2 - 3x + 2 is a quadratic polynomial function as the highest exponent of x is 2. In contrast, g(x) = 2x - 4 is a linear polynomial function as the highest exponent is 1.
  • The general shape, behavior, and properties such as continuity and differentiability of polynomial functions make them very manageable and predictable within mathematics.
  • Their evaluation, as seen in this exercise, follows clear algebraic rules, and working with them offers excellent practice for understanding more complicated functions.
Grasping the structure and characteristics of polynomials is vital for students as they are used extensively in both theoretical and applied mathematics.

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Most popular questions from this chapter

Solve by completing the square or by using the quadratic formula. $$x^{2}-3 x+5$$

Use a graphing utility. Graph: \(f(x)=x^{2}-2|x|-3\)

The notation \(\left.f(x)\right|_{a} ^{b}\) is used to denote the difference \(f(b)-f(a) .\) That is, $$\left.f(x)\right|_{a} ^{b}=f(b)-f(a)$$ Evaluate \(\left.f(x)\right|_{0} ^{b}\) for the given function \(f\) and the indicated values of \(a\) and \(b\). $$f(x)=2 x^{3}-3 x^{2}-x ;\left.f(x)\right|_{0} ^{2}$$

Use a graphing utility. Graph: \(f(x)=\left|x^{2}-1\right|-|x-2|\)

If \(y=x^{2}-3 x+2,\) find \(x\) when \(y=0 .[1.1]\)
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