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Chapter 1: Problem 15

Evaluate the indicated function for \(f(x)=x^{2}+1\) and \(g(x)=x-4\). $$(f-g)(0)$$

Short Answer

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Step by step solution

01

Determine the difference function

To start with, we can find the difference function \(f(x) - g(x)\). Using the given functions, \(f(x) - g(x)\) = \((x^{2} + 1) - (x - 4)\) = \(x^{2} - x + 5\).
02

Evaluate the function at the given point

After determining the difference function, we have \(f(x) - g(x) = x^{2} - x + 5\). Now we evaluate this function at \(x = 0\) to find \((f - g)(0)\). Since \(x = 0\), our result is \(0^{2} - 0 + 5 = 5\). Therefore, \((f-g)(0) = 5\)

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Key Concepts

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

Difference of Functions
Understanding the difference of functions is a fundamental concept in algebra. It involves subtracting one function from another. For any two functions, say \(f(x)\) and \(g(x)\), their difference is represented by \((f-g)(x)\).
  • To calculate this, you subtract the entire expression of \(g(x)\) from \(f(x)\).
  • The most common mistake students make is not distributing the negative sign across \(g(x)\).
For example, if \(f(x) = x^2 + 1\) and \(g(x) = x - 4\), the difference would be \(f(x) - g(x) = (x^2 + 1) - (x - 4)\). Remember to distribute the subtraction sign, which results in \(x^2 - x + 5\). Once you have this expression, you can evaluate it at specific values of \(x\) as needed.
Polynomial Functions
Polynomial functions are expressions that involve sums of constants and variables raised to whole number powers. They are among the most common functions used in algebra.
  • Each term in a polynomial is called a 'monomial', and is typically in the form of \(a_n \cdot x^n\), where \(a_n\) is a constant coefficient, and \(n\) is a non-negative integer.
  • The degree of the polynomial is the highest power of the variable in the expression.
Take the function \(f(x) = x^2 + 1\), for example. It is a polynomial function with:
  • The degree of 2, since \(x^2\) is the term with the highest exponent.
  • Two terms: \(x^2\) and \(+1\).
Polynomial functions can be easily added, subtracted, multiplied, and evaluated at specific points. They form the basis for a wide range of useful mathematical models.
Function Operations
Function operations involve combining two or more functions through addition, subtraction, multiplication, or division. This is a key concept in algebra that shows how different mathematical functions interact.
  • Addition and subtraction of functions are straightforward鈥� you simply add or subtract the corresponding terms.
  • For multiplication, you apply the distributive property.
  • Division is typically not as straightforward, especially if the denominator is zero at any point.
In our example, we handled a subtraction operation between functions \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). We derived the difference as \(x^2 - x + 5\). This subtraction shows how specific values from one function influence another.Evaluating this operation at specific points, like at \(x = 0\), is often the final step in a problem, revealing the value of the new function at that point.

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Most popular questions from this chapter

The height \(y\) (in feet) of a baseball thrown by a child is $$y=-\frac{1}{10} x^{2}+3 x+6$$ where \(x\) is the horizontal distance (in feet) from where the ball was thrown. Will the ball fly over the head of another child 30 feet away trying to catch the ball? (Assume that the child who is trying to catch the ball holds a baseball glove at a height of 5 feet.)

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=\frac{5}{6}-\frac{2}{3} x$$

Find the difference quotient and simplify your Answer: $$f(x)=x^{3}+3 x, \quad \frac{f(x+h)-f(x)}{h}, \quad h \neq 0$$

Beam Load The maximum load that can be safely supported by a horizontal beam varies jointly as the width of the beam and the square of its depth and inversely as the length of the beam. Determine the changes in the maximum safe load under the following conditions. A. The width and length of the beam are doubled. B. The width and depth of the beam are doubled.

The cost of sending an overnight package from New York to Atlanta is 26.10 dollars for a package weighing up to, but not including, 1 pound and 4.35 dollars for each additional pound or portion of a pound. (a) Use the greatest integer function to create a model for the cost \(C\) of overnight delivery of a package weighing \(x\) pounds, \(x>0\). (b) Sketch the graph of the function.
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