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MATH / Consider the two functions f(x)=x²+2bx+1 and g(x)=2a(x+b), where the variables x and the constants a and b are real numbers. Each such pair of constants a and b may be considered a…

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problem

Consider the two functions f(x)=x2+2bx+1f(x) = x^2 + 2bx + 1 and g(x)=2a(x+b),g(x) = 2a(x + b), where the variables xx and the constants aa and bb are real numbers. Each such pair of constants aa and bb may be considered as a point (a,b)(a,b) in an abab-plane. Let SS be the set of points (a,b)(a,b) for which the graphs of y=f(x)y = f(x) and y=g(x)y = g(x) do not intersect (in the xyxy-plane). Find the area of S.S.
Plain-text mathematical notation (without MathML)
Consider the two functions f(x)=x²+2bx+1 and g(x)=2a(x+b), where the variables x and the constants a and b are real numbers.  Each such pair of constants a and b may be considered as a point (a,b) in an ab-plane.  Let S be the set of points (a,b) for which the graphs of y=f(x) and y=g(x) do not intersect (in the xy-plane).  Find the area of S.
Original LaTeX notation
Consider the two functions $f(x) = x^2 + 2bx + 1$ and $g(x) = 2a(x + b),$ where the variables $x$ and the constants $a$ and $b$ are real numbers.  Each such pair of constants $a$ and $b$ may be considered as a point $(a,b)$ in an $ab$-plane.  Let $S$ be the set of points $(a,b)$ for which the graphs of $y = f(x)$ and $y = g(x)$ do not intersect (in the $xy$-plane).  Find the area of $S.$

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
f9f49fd9-01d9-59ee-b961-cf057a7d7a2c
task_key
intermediate~5falgebra--test--f9f49fd9-01d9-59ee-b961-cf057a7d7a2c
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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