Question

Determine whether the function is one-to-one.
f(x)=(x-5)^3

Answer 1

Horizontal Line Test
• If some horizontal line intersects the graph of the function more than once,
then the function is not one-to-one.
• If no horizontal line intersects the graph of the function more than once,
then the function is one-to-one.
What are One-To-One Functions? Algebraic Test
Definition 1. A function f is said to be one-to-one (or injective) if
f(x1) = f(x2) implies x1 = x2.
Lemma 2. The function f is one-to-one if and only if
8x1, 8x2, x1 6= x2 implies f(x1) 6= f(x2).
Examples and Counter-Examples
Examples 3. • f(x) = 3x − 5 is 1-to-1.
• f(x) = x2 is not 1-to-1.
• f(x) = x3 is 1-to-1.
• f(x) = 1
x is 1-to-1.
• f(x) = xn − x, n > 0, is not 1-to-1.
Proof. • f(x1) = f(x2) ) 3x1 − 5 = 3x2 − 5 ) x1 = x2. In
general, f(x) = ax − b, a 6= 0, is 1-to-1.
• f(1) = (1)2 = 1 = (−1)2 = f(−1). In general, f(x) = xn, n even, is not
1-to-1.
• f(x1) = f(x2) ) x31
= x32
) x1 = x2. In general, f(x) = xn, n
odd, is 1-to-1.
• f(x1) = f(x2) ) 1
x1
= 1
x2
) x1 = x2. In general, f(x) = x−n, n
odd, is 1-to-1.
• f(0) = 0n − 0 = 0 = (1)n − 1 = f(1). In general, 1-to-1 of f and g does
not always imply 1-to-1 of f + g.
Properties of One-To-One Functions
Properties
Properties
If f and g are one-to-one, then f g is one-to-one.
Proof. f g(x1) = f g(x2) ) f(g(x1)) = f(g(x2)) ) g(x1) =
g(x2) ) x1 = x2.
Examples 4. • f(x) = 3x3 − 5 is one-to-one, since f = g u where g(u) =
3u − 5 and u(x) = x3 are one-to-one.
• f(x) = (3x − 5)3 is one-to-one, since f = g u where g(u) = u3 and
u(x) = 3x − 5 are one-to-one.
• f(x) = 1
3x3−5 is one-to-one, since f = g u where g(u) = 1
u and u(x) =
3x3 − 5 are one-to-one.
2

Answer 2

hope I helped

Answer 3

Yes many thanks

Answer 4

no problem :)