#function composition

General Mathematics 3

If f: N→N, g: N→N and h: N→R is defined f(x) = 3x - 5, g(y) = 6y2 and h(z) = tan⁡z, find ho(gof).

Understanding this is easy: using formula we get tan⁡(6(3x - 5)²). Calculation gives this directly, check steps: apply rule, substitute values, simplify to get tan⁡(6(3x - 5)²).

Ref: NCERT Mathematics Class XI and XII

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General Mathematics 3

If f : R → R is given by f(x) = (5 + x⁴)¹/⁴, then fοf(x) is _______

Notice that using formula we get (10 + x⁴)¹/⁴. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get (10 + x⁴)¹/⁴.

Ref: NCERT Mathematics Class XI and XII

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General Mathematics 2

If f : R→R, g(x) = 3 x 2 + 7 and f(x) = √x, then gοf(x) is equal to _______

Applying the basic concept, solving fully: apply relevant formula, put numbers, compute. Result comes as 3x + 7. Easy verification shows it satisfies condition.

Ref: NCERT Mathematics Class XI and XII

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Sets, Relations and Functions

If f(x): R → Z and f(x) = [x], where [.] denotes smallest integer function and g(x): Z → (0,∞) and g(x) = 5x then find t

We know that using formula we get 5. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get 5.

Ref: NCERT Mathematics Class XI — Sets, Relations and Functions

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Sets, Relations and Functions

Let f(x) = px + q and g(x) = mx + n. Then f (g(x)) = g (f(x)) is equivalent to

Notice that after simplifying, f(n) = g(q) is left. The method is simple: formula → substitution → calculation → answer.

Ref: NCERT Mathematics Class XI — Sets, Relations and Functions

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Sets, Relations and Functions

Let f(x) = x - 6 and g(x) = x² - 2x + 3 where x is a positive integer. Find x at which g(f(x)) + f(g(x) is minimum?

Think of it like this: we verify 4 by full calculation: use standard identity, substitute, simplify step by step to reach 4.

Ref: NCERT Mathematics Class XI — Sets, Relations and Functions

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Differential Calculus

If f(x) is an increasing function and g(x) is a decreasing function such that gof(x) is defined, then gof(x) will be

Simply put, after simplifying, gof(x) is decreasing function is left. The method is simple: formula → substitution → calculation → answer.

Ref: NCERT Mathematics Class XII — Applications of Derivatives

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