Show that the function f : R R given by f(x)=3 x+4 is injective. - Vidyadip

Question

Show that the function $f : R \rightarrow R$ given by $f(x)=3 x+4$ is injective.

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Answer

To prove a function is injective (one-to-one), we assume $f\left(x_1\right)=f\left(x_2\right)$ and show that $x_1=x_2$.
Let $x_1, x_2 \in R$ such that $f\left(x_1\right)=f\left(x_2\right)$.
Substitute the function definition: $3 x_1+4=3 x_2+4$.
Subtract 4 from both sides: $3 x_1=3 x_2$.
Divide by 3: $x_1=x_2$.
Since $f\left(x_1\right)=f\left(x_2\right) \Longrightarrow x_1=x_2$, the function $f$ is injective.

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