Question

find the range of f(x) = 5/3-x^2​

Answer 1

Examples with Solutions

Example 1

Find the Range of function f defined by  

f (x) = - 3

Solution to Example 1

The given function has a constant value 3 and therefore the range is the set  

{3}  

Example 2

Find the Range of function f defined by  

f (x) = 4 x + 5

Solution to Example 2

Assuming that the domain of the given function is the set of all real numbers R, so that the variable x takes all values in the interval  

(-∞ , +∞)  

If x takes all values in the interval (-∞ , +∞) then 4 x + 5 takes all values in the interval (-∞ , +∞) and the range of the given function is given by the interval (-∞ , +∞)

Example 3

Find the Range of function f defined by  

f (x) = x 2 + 5

Solution to Example 3

Assuming that the domain of the given function is R meaning that x takes all values in the interval (-∞ , +∞) which means that x 2 is either zero or positive. Hence we can write the following inequality  

x 2 ≥ 0  

Add 5 to both sides of the inequality to obtain the inequality  

x 2 + 5 ≥ = 0 + 5 or f(x) ≥ 5  

The range of f (x) = x 2 + 5 is given by the interval

Answer 2

The range of the function $f(x) = \frac{5}{3}-x^2$ is (- ∞, $\frac{5}{3}$ ].

Given,

A function $f(x) = \frac{5}{3}-x^2$.

To Find,

The range of the function.

Solution,

The method of finding the range of the function is as follows -

We know that the range of the function $g(x)=x^2$ is [ 0, ∞ ).

So the range of the function $g_1(x)=-x^2$ will be (- ∞, 0 ].

Now we can observe if we add $\frac{5}{3}$ to $g_1(x)$ we will get f(x) i.e. $f(x)=\frac{5}3}+g_1(x)$.

So the range of f(x) will be  (- ∞, $0+\frac{5}{3}$ ] = (- ∞, $\frac{5}{3}$ ].

Hence, the range of the function $f(x) = \frac{5}{3}-x^2$ is  (- ∞, $\frac{5}{3}$ ].

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