Question

Find the domain and range of the following function.

f(x) = 7x² + 4x − 1

Answer 1

f(x) = 7x² + 4x − 1

f is defined for all x

∴ Domain of f = R (i.e. the set of real numbers)

7x² + 4x − 1

= ${\displaystyle 7(x^2+\frac{4}{7}x)-1}$

= ${\displaystyle 7{(x+\frac{2}{7})}^2-1-\frac{4}{7}}$

= ${\displaystyle 7{(x+\frac{2}{7})}^2-\frac{11}{7}\ge -\frac{11}{7}}$

∴ Range of f = ${\displaystyle [-\frac{11}{7},\infty )}$

Answer 2

f(x) = 7x² + 4x - 1

f(x) is defined for all x ∈ R

∴ Domain = R

7x² + 4x − 1

${\displaystyle =7[x^2+\frac{4}{7}x]-1}$

${\displaystyle =7[x^2+\frac{4x}{7}+\frac{4}{49}]-\frac{28}{49}-1}$

${\displaystyle =7{(x+\frac{2}{7})}^2-\frac{77}{49}}$

${\displaystyle =7{(x+\frac{2}{7})}^2-\frac{11}{7}}$

∴  ${\displaystyle 7{(x+\frac{2}{7})}^2\ge 0 \text{for all x ∈ R}}$

∴  ${\displaystyle 7{(x+\frac{2}{7})}^2-\frac{11}{7}\ge 0-\frac{11}{7}\text{for all x ∈ R}}$

∴ ${\displaystyle f\left(x\right)\ge -\frac{11}{7}\text{for all x ∈ R}}$

∴ Range = ${\displaystyle [-\frac{11}{7},\infty )}$

∴ Domain = R, Range = ${\displaystyle [-\frac{11}{7},\infty ).}$