Suppose g(x)=2 x+1 and h(x)=4 x^2+4 x+5 and h(x)=(f ∘g)(x)....

Suppose $g(x)=2 x+1$ and $h(x)=4 x^{2}+4 x+5$ and $h(x)=(f \circ g)(x)$. The area enclosed by the graph of the function $y=f(x)$ and the pair of tangents drawn to it from the origin, is
(a) $8 / 3$
(b) $16 / 3$
(c) $32 / 3$
(d) None of these

Key Concepts

Area Enclosed by Curves and Tangents

Calculating the area enclosed by a curve and its tangents involves integrating the difference between the functions over the interval defined by their points of intersection. This application of definite integration is fundamental in calculus for finding areas between curves, and it often requires determining the limits of integration by solving for points where the functions intersect.

Condition for Tangency

The condition for tangency in calculus is that the line and the curve meet at exactly one point, which mathematically translates to the discriminant of the resulting equation (when equating the line and curve) being zero. This condition is used to ensure that a line touches the curve at only one point and does not cross it.

Tangents from an External Point

Drawing tangents from a point external to a curve is a classical problem in geometry and calculus. This concept involves determining the lines that touch the curve at exactly one point while passing through the given external point. This requires setting up an equation of a line through the external point and imposing the condition that it is tangent to the curve.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. In general, if h(x) = (f ? g)(x), then h(x) equals f(g(x)). Understanding how to decompose a composite function is crucial in many areas of mathematics, as it allows one to determine unknown components by working backwards from known functions.

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