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Chapter 1: Problem 15

Given \(f(x)=x^{2}+2 x+3\), find \(\quad\) a. \(f(0) \quad\) b. \(f(1)\) c. \(f(-1)\) d. \(f(-3)\) e. \(f(a) \quad\) f. \(f(x+h)\)

Short Answer

Expert verified
a. \(f(0) = 3\) b. \(f(1) = 6\) c. \(f(-1) = 2\) d. \(f(-3) = 6\) e. \(f(a) = a^2 + 2a + 3\) f. \(f(x+h) = x^2 + 2xh + h^2 + 2x + 2h + 3\)

Step by step solution

01

Evaluate \(f(0)\)

To find \(f(0)\), we'll substitute \(x=0\) in the function \(f(x)\). $f(0) = (0)^{2} + 2(0) + 3 = 0 + 0 + 3 = 3$ #b. f(1)#
02

Evaluate \(f(1)\)

To find \(f(1)\), we'll substitute \(x=1\) in the function \(f(x)\). $f(1) = (1)^{2} + 2(1) + 3 = 1 + 2 + 3 = 6$ #c. f(-1)#
03

Evaluate \(f(-1)\)

To find \(f(-1)\), we'll substitute \(x=-1\) in the function \(f(x)\). $f(-1) = (-1)^{2} + 2(-1) + 3 = 1 - 2 + 3 = 2$ #d. f(-3)#
04

Evaluate \(f(-3)\)

To find \(f(-3)\), we'll substitute \(x=-3\) in the function \(f(x)\). $f(-3) = (-3)^{2} + 2(-3) + 3 = 9 - 6 + 3 = 6$ #e. f(a)#
05

Evaluate \(f(a)\)

To find \(f(a)\), we'll substitute \(x=a\) in the function \(f(x)\). \(f(a) = a^{2} + 2a + 3\) For a general input \(a\), the function value is \(f(a) = a^{2} + 2a + 3\). #f. f(x+h)#
06

Evaluate \(f(x+h)\)

To find \(f(x+h)\), we'll substitute \(x=x+h\) in the function \(f(x)\). $f(x+h) = (x+h)^{2} + 2(x+h) + 3 = (x^{2} + 2xh + h^{2}) + 2(x + h) + 3 = x^{2} + 2xh + h^{2} + 2x + 2h + 3$ For an input of \(x+h\), the function value is \(f(x+h) = x^{2} + 2xh + h^{2} + 2x + 2h + 3\).

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Key Concepts

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

Function Evaluation
Function evaluation is a fundamental concept in mathematics where you substitute a specific value or expression for the variable in a function. It helps in determining the output based on different inputs.
Function evaluation is particularly important in polynomials, like quadratic functions. Here's how it works:
  • Choose the value you want to substitute for the variable.
  • Replace the variable with the chosen value in the function expression.
  • Simplify the expression to find the resulting output.
This method helps in verifying how a function behaves for specific inputs, which can be numbers or even other expressions like another variable. Through evaluation, you get concrete outputs that help understand the overall behavior of the function.
Substitution Method
The substitution method is a key technique used in evaluating functions and solving equations. It involves replacing a variable within an expression or equation with a specific value or another expression, allowing for simplification and easier computation.
Steps to approach the substitution method include:
  • Identify the variable to be substituted and the value you want to replace it with.
  • Substitute the identified value into the expression in place of the variable.
  • Simplify the resulting expression to obtain a simplified form or a numerical result.
This method simplifies the process of evaluating functions at specific points, making it particularly useful with polynomial expressions, like quadratics, where precision is key to understanding its properties.
Quadratic Functions
Quadratic functions are a type of polynomial function characterized by the highest degree being two. This means the variable is raised to the power of two, and the standard form of a quadratic function is given by \[ f(x) = ax^2 + bx + c \]
Where:
  • \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).
  • The expression involves terms like \( x^2 \), \( x \), and a constant term \( c \) which collectively determine its shape and position on a graph.
Quadratic functions create parabolic graphs. They are symmetrical and have certain defining features like:
  • Vertex: The highest or lowest point of the parabola.
  • Axis of symmetry: A vertical line that divides the parabola into two symmetrical halves.
  • Roots: The points where the function crosses the x-axis, which can be found by solving the equation \( ax^2 + bx + c = 0 \).
These functions are widely used in various fields such as physics, engineering, and economics for modeling situations where relationships between variables are represented in a squared, parabolic manner.

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