Problem 9 - Precalculus with Limits

Question

In Exercises 1-9, match each function with its name. \(f(x) = ax + b\) (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (g) greatest integer function (h) reciprocal function (i) identity function

Step-by-Step Solution

Answer

The function \(f(x) = ax + b\) corresponds to a linear function.

Step 1: Problem

In Exercises 1-9, match each function with its name.

\(f(x) = ax + b\)

(a) squaring function
(b) square root function
(c) cubic function
(d) linear function
(e) constant function
(f) absolute value function
(g) greatest integer function
(h) reciprocal function
(i) identity function

Step 2: Notation

\(|x-a|\) = distance from \(x\) to \(a\).

Step 3: Expression

The function \(f(x) = ax + b\) corresponds to a linear function.

Key Concepts

Function Types

Functions are mathematical expressions that show the relationship between inputs and outputs. They come in various types, each with its own distinct form and graph.
Here are the most common types you might encounter:

• Linear Function: Represents a constant rate of change, and its equation is typically in the form of \(f(x) = ax + b\), where \(a\) and \(b\) are constants. The graph of a linear function is a straight line.
• Constant Function: A special case of a linear function where the slope \(a\) is zero. The output remains the same no matter the input, represented as \(f(x) = b\).
• Quadratic Function: Takes the form of \(f(x) = ax^2 + bx + c\) and graphs as a parabola.
• Cubic Function: Given by \(f(x) = ax^3 + bx^2 + cx + d\), which can produce more complex curves.
• Square Root Function: Involves the square root, \(f(x) = \sqrt{x}\), usually forming a curve that begins at the origin and rises steadily.
• Absolute Value Function: Shown as \(f(x) = |x|\), it forms a V-shape where all values are non-negative.
• Reciprocal Function: With the equation \(f(x) = \frac{1}{x}\), creating a hyperbola as its graph.

Recognizing these various function types is key to understanding and solving algebraic equations effectively.

Graphing Functions

Graphing functions is an efficient way to visualize the relationship between variables. The graph of a function provides insight into its behavior and characteristics.
Here's how to graph common functions:

• For a Linear Function: Use the slope-intercept form, \(y = mx + c\), to find the y-intercept (where the line crosses the y-axis) and use the slope to determine the direction of the line. Plot these points and draw a straight line through them.
• Quadratic Function: Plot the vertex and a few points on either side symmetrically. The curve will be parabolic, opening upwards or downwards depending on the sign of \(a\).
• Absolute Value Function: Plot the vertex at the origin, with arms opening upwards forming a 'V' shape. The slope will affect the steepness of the lines forming the 'V'.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we graph mathematical functions. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis) that intersect at a point called the origin.
Understanding the coordinate plane is fundamental for graphing and interpreting functions:

• The X-Axis: Represents the input or independent variable. It's horizontal, and numbers increase from left to right.
• The Y-Axis: Represents the output or dependent variable. It's vertical, with values increasing upwards.
• The Origin: Denoted as \((0, 0)\), where both axes intersect. It's the starting point for measuring coordinates.
• Quadrants: The coordinate plane is divided into four quadrants, each with positive and negative values. In the first quadrant, both x and y are positive, while they have varying signs in the other quadrants.

The coordinate plane enables us to plot points, lines, and curves, revealing the shape and direction of functions. With a proper understanding of this plane, interpreting and drawing functions becomes much smoother.

Previous

Problem 9

Next

Problem 9

Other exercises in this chapter

Problem 9

In Exercises 7-14, find the inverse function of \(f\) informally. Verify that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). \(f(x) = x + 9\)

View solution

Problem 9

In Exercises 9-16, find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((fg)(x)\), and (d) \((f/g)(x). What is the domain of \)f/g\(? \)f(x) = x + 2\(, \)g(x) = x - 2

View solution

Problem 9

In Exercises 7-14, determine whether each point lies on the graph of the equation. \( y = x^2 - 3x + 2 \) (a) \( (2, 0) \) (b) \( (-2, 8) \)

View solution

Problem 9

In Exercises 7-10, plot the points in the Cartesian plane. \( (3, 8) \), \( (0.5, -1) \), \( (5, -6) \), \( (-2, 2.5) \)

View solution