First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightFunctions And Graphschevron_rightExercise 2.1

Solution

Given $f(x)=3x-5$ .

(i) Domain and range

A linear function is defined for all real $x$ and produces all real values.

\boxed{D_f=\mathbb{R},\quad R_f=\mathbb{R}}

(ii) One-to-one

Assume $f(x_1)=f(x_2)$ :

3x_1-5=3x_2-5\Rightarrow 3x_1=3x_2\Rightarrow x_1=x_2

So $f$ is one-to-one.

\boxed{f\ \text{is one-to-one}}

(iii) Onto (co-domain $\mathbb{R}$ )

Let $y\in\mathbb{R}$ . Solve $y=3x-5$ :

y+5=3x\Rightarrow x=\frac{y+5}{3}

Since $\frac{y+5}{3}\in\mathbb{R}$ , every real $y$ has a preimage, so $f$ is onto.

\boxed{f\ \text{is onto }\mathbb{R}}

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