Practice Flashcards
What does (f∘g)(x) mean?
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All Flashcards in Topic 2.5
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2.5.19 cards
What does (f∘g)(x) mean?
f(g(x)) — apply the inner function g first, then f. Read right-to-left.
Does f∘g equal g∘f?
Not in general — order matters; the inner function changes the result.
How do you evaluate (f∘g)(a) at a number?
Compute g(a) first, then put that value into f. Work inside-out.
How do you form the composite expression (f∘g)(x)?
Replace every x in f with the whole expression g(x) (in brackets), then simplify.
f(x) = 2x + 1, g(x) = x². Find (f∘g)(x).
f(x²) = 2x² + 1.
f(x) = 2x + 1, g(x) = x². Find (g∘f)(x).
g(2x + 1) = (2x + 1)² = 4x² + 4x + 1.
How do you solve a composite equation like (f∘g)(x) = k?
Form the composite expression, set it equal to k, and solve for x.
How do you find f given g and (f∘g)(x)?
Compose with the unknown f, then match coefficients to the given result.
Common composite mistake?
Doing the functions in the wrong order, or forgetting brackets when substituting (e.g. (2x+1)²).
2.5.28 cards
How do you find f⁻¹ algebraically?
Write y = f(x), swap x and y, solve for y. (Geometrically, reflect in y = x.)
How do you invert a rational function?
Swap x and y, multiply up to clear the fraction, gather the y-terms, factor out y, then divide.
Find the inverse of f(x) = 5x − 2.
x = 5y − 2 ⇒ y = (x + 2)/5, so f⁻¹(x) = (x + 2)/5.
What composition check confirms an inverse?
f(f⁻¹(x)) = x (and f⁻¹(f(x)) = x) — they undo each other.
How do the domain and range of f⁻¹ relate to f?
They swap: domain of f⁻¹ = range of f; range of f⁻¹ = domain of f.
Why does x² need a restricted domain to have an inverse?
x² isn't one-to-one over all x; restricting to x ≥ 0 makes it invertible, giving f⁻¹(x) = √x.
Inverse of f(x) = (2x + 1)/(x − 3)?
Swap, multiply up, gather y: f⁻¹(x) = (3x + 1)/(x − 2).
Is f⁻¹ the same as 1/f?
No — f⁻¹ is the inverse function; 1/f is the reciprocal.
2.5.35 cards
How do you read f(a) off a graph?
Go up from x = a to the curve, then across to the y-axis.
How do you read (f∘f)(a) off a graph?
Read f(a), then read f of that result — two read-offs, inside first.
How do you read f⁻¹(b) off the graph of f?
Start at y = b on the y-axis, go across to the curve, then down to the x-axis.
How do you sketch y = f⁻¹(x) from y = f(x)?
Reflect the graph in the line y = x; every point (a, b) becomes (b, a).
What happens to intercepts under f → f⁻¹?
They swap: a y-intercept (0, k) becomes an x-intercept (k, 0), and vice versa.
Topic 2.5 study notes
Full notes & explanations for Composite & inverse functions
Math AA exam skills
Paper structures, command terms & tips
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