Textbook Question
Functions
In Exercises 1–6, find the domain and range of each function.
G(t) = 2/(t² − 16)
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0. Functions
Introduction to Functions
2:47 minutesProblem 1.1.12
Textbook Question
Finding Formulas for Functions
A point P in the first quadrant lies on the graph of the function f(x) = √x. Express the coordinates of P as functions of the slope of the line joining P to the origin.
Verified step by step guidance1
Identify the coordinates of point P on the graph of f(x) = √x. Since P lies on this graph, its coordinates can be expressed as (a, √a) for some positive value a.
Determine the slope of the line joining the origin (0, 0) to the point P (a, √a). The slope m is given by the formula m = (√a - 0) / (a - 0) = √a / a.
Express a in terms of the slope m. From the equation m = √a / a, we can rearrange to find a in terms of m: m = 1/√a, which implies √a = 1/m.
Solve for a by squaring both sides of the equation √a = 1/m to get a = 1/m².
Substitute a = 1/m² back into the coordinates of P. The x-coordinate is a = 1/m², and the y-coordinate is √a = 1/m. Therefore, the coordinates of P as functions of the slope m are (1/m², 1/m).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Slope of a Line
The slope of a line measures its steepness and is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points. For a line joining the origin (0,0) to a point P(x, f(x)), the slope can be expressed as f(x)/x. Understanding this concept is crucial for relating the coordinates of point P to the slope.
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Function Representation
A function represents a relationship between inputs and outputs, typically expressed as f(x). In this case, f(x) = √x defines the output for any given input x. Recognizing how to manipulate and express functions is essential for deriving the coordinates of point P in terms of the slope.
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Coordinate Geometry
Coordinate geometry involves the study of geometric figures using a coordinate system. In this context, the coordinates of point P are expressed as (x, √x). Understanding how to work with coordinates in the Cartesian plane is vital for solving problems related to slopes and functions.
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