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Chapter 1: Problem 30

Use the given function \(f\) to find \(f(0)\) and solve \(f(x)=0\) $$f(x)=x^{2}-x-12$$

Short Answer

Expert verified
f(0) = -12; solutions are x = 4 and x = -3.

Step by step solution

01

Find f(0)

To find the value of the function at zero, substitute \(x = 0\) into the function. This gives us\[f(0) = (0)^2 - 0 - 12 = -12.\]Therefore, \(f(0) = -12.\)
02

Solve f(x) = 0 by Factoring

To solve \(f(x) = x^2 - x - 12 = 0\), we factor the quadratic expression. First, look for two numbers whose product is -12 and whose sum is -1. The numbers -4 and 3 satisfy this condition. Thus, we factor the quadratic as\[x^2 - x - 12 = (x - 4)(x + 3).\]
03

Solve the Factored Equation

With the equation factored, solve for \(x\) in\[(x - 4)(x + 3) = 0.\]This gives us two solutions: \(x - 4 = 0\) or \(x + 3 = 0\). Solving these, we find:- \(x = 4\)- \(x = -3\).

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Key Concepts

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

Factoring Quadratics
Factoring quadratics is a fundamental skill when working with quadratic functions. Quadratic functions have the general form: \[ f(x) = ax^2 + bx + c \]
Where \(a\), \(b\), and \(c\) are constants. Factoring involves rewriting this polynomial as a product of two binomials.
  • The goal is to find two numbers that multiply to give \( ac \) (in our example \(a = 1\) so just \( c \), which is \(-12\)) and add to give \( b \) (which is \(-1\) in our example).
  • These numbers become the constants in our binomials.
For the function \( f(x) = x^2 - x - 12 \), the numbers \(-4\) and \(3\) meet the criteria as \((-4) \times 3 = -12\) and \((-4) + 3 = -1\). From there, the quadratic can be factored as:\[ f(x) = (x - 4)(x + 3) \]
This transformation allows for easier manipulation and solution of the equation.
Solving Equations
Once a quadratic like \( f(x) = x^2 - x - 12 \) is factored into a form such as \((x - 4)(x + 3) = 0\), solving the equation becomes straightforward. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. Thus, to find the solutions to \( f(x) = 0 \), set each factor to zero:
  • \( x - 4 = 0 \)
  • \( x + 3 = 0 \)
Solving each of these simple equations gives:
  • From \( x - 4 = 0 \), add \(4\) to both sides to get \( x = 4 \).
  • From \( x + 3 = 0 \), subtract \(3\) from both sides to get \( x = -3 \).
Hence, the roots or solutions of the equation are \( x = 4 \) and \( x = -3 \). This means the graph of the quadratic function will cross the x-axis at these points.
Function Evaluation
Evaluating a function at a specific point helps us understand its behavior at that point. To evaluate a quadratic function like \( f(x) = x^2 - x - 12 \) at a particular value of \( x \), substitute that value into the function.For example, to evaluate \( f(0) \), substitute \( x = 0 \) into the function:\[ f(0) = (0)^2 - 0 - 12 = -12 \]
This result tells us that when \( x = 0 \), the function's output is \(-12\). Such evaluation is particularly useful in finding the y-intercept of the graph of the function, providing valuable information about one of the key points on a quadratic curve.

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Most popular questions from this chapter

In Exercises \(21-45,\) find and simplify the difference quotient \(\frac{f(x+h)-f(x)}{h}\) for the given function. $$ f(x)=\sqrt{x-9} $$

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