What is the range of f(x)=(a^2+5)/3 ?

The range of f(x)=(a^2+5)/3 is f(x)>= 5/3

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range of f(x)=(a^2+5)/3

range

f (x) = \frac{a ^{2} + 5}{3}

f (x) \geq \frac{5}{3}

Interval Notation

[\frac{5}{3}, \infty)

Solution steps

Rewrite as

\frac{a ^{2} + 5}{3} = y

Multiply both sides by

3

a^{2} + 5 = 3 y

Move

3 y

to the left side

a^{2} + 5 - 3 y = 0

For a quadratic equation of the form

a x^{2} + b x + c = 0

the discriminant is

b^{2} - 4 a c

For

a = 1, b = 0, c = 5 - 3 y : 0^{2} - 4 \cdot 1 (5 - 3 y)

0^{2} - 4 \cdot 1 \cdot (5 - 3 y)

Expand

0^{2} - 4 \cdot 1 \cdot (5 - 3 y) : 12 y - 20

The range is the set of y for which the discriminant is greater or equal to zero

12 y - 20 \geq 0 : y \geq \frac{5}{3}

Check if the range interval endpoints are included

y = \frac{5}{3} :

Included

Therefore the range is

f (x) \geq \frac{5}{3}

in t erce pt s f (x) = (x + 4)^{2} - 7

in v erse f (x) = \frac{x - 8}{x + 8}

p er p e n d i c u l a r y = - \frac{1}{3} x + 9

in v erse f (x) = 3 x + 2

in v erse y = \frac{4}{x + 3}