Show that the function f- - given by fx-x3 is injective-

F: R → R is given as f(x) = x3. Suppose f(x) = f(y), where x, y ∈ R. ⇒ x3 = y3 … (1) Now, we need to show that x = y. Suppose x ≠ y, their cubes will also not ...

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Standard XII

Mathematics

Properties of Determinants

Show that the...

Question

Show that the function f: R → R given by f(x) = x³ is injective.

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Solution

f: R → R is given as f(x) = x³.

Suppose f(x) = f(y), where x, y ∈ R.

⇒ x³ = y³ … (1)

Now, we need to show that x = y.

Suppose x ≠ y, their cubes will also not be equal.

x³ ≠ y³

However, this will be a contradiction to (1).

∴ x = y

Hence, f is injective.

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Similar questions

Q. Show that the function

f : R \to R

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f (x) = x^{3}

is injective

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(iii) f: R → R given by f(x) = x²

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(v) f: Z → Z given by f(x) = x³

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Show that the function f: R → R given by f(x) = x3 is injective.

f: R → R is given as f(x) = x3. Suppose f(x) = f(y), where x, y ∈ R. ⇒ x3 = y3 … (1) Now, we need to show that x = y. Suppose x ≠ y, their cubes will also not be equal. x3 ≠ y3 However, this will be a contradiction to (1). ∴ x = y Hence, f is injective.

Show that the function f: R → R given by f(x) = x³ is injective.

f: R → R is given as f(x) = x³.

Suppose f(x) = f(y), where x, y ∈ R.

⇒ x³ = y³ … (1)

Now, we need to show that x = y.

Suppose x ≠ y, their cubes will also not be equal.

x³ ≠ y³

However, this will be a contradiction to (1).

∴ x = y

Hence, f is injective.