Q.Show that the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 5x - 7$ is one-one and onto.

One-one: suppose $f(x_1)=f(x_2)$. Then $5x_1-7=5x_2-7 \Rightarrow 5x_1=5x_2 \Rightarrow x_1=x_2$. So $f$ is one-one.

Onto: for any $y\in\mathbb{R}$, solve $5x-7=y \Rightarrow x=\dfrac{y+7}{5}$, a real number for every real $y$, and $f\left(\dfrac{y+7}{5}\right)=y$. So $f$ is onto.

Assume equal outputs and solve for equal inputs to show injectivity; take an arbitrary output and solve backward for a valid input to show surjectivity.

• Sign errors moving the constant $-7$ across the equation.
• Forgetting to explicitly confirm $x=\frac{y+7}{5}$ is a real number for every real $y$.
• Treating 'linear function' as automatically bijective without actually writing out the check.