Question

What is the range of a piecewise function g of x, where part 1 is x squared minus 5 for x less than negative 3 and part 2 is 2 times x for x greater than or equal to negative 3.? (–[infinity], [infinity]) [–6, [infinity]) [–5, [infinity]) [5, [infinity])

Answer 1

The range of a piecewise function g(x) for x < -3 and $x\geq-3$ is ( -6, ∞ ).

     

Given function,

$g(x)=\left \{ {{x^2-5 ~for~x ~ < ~-3 } \atop {2x~for~x~\geq-3}} \right.$
We have to find the range of this function.

If we take x < -3 we have to take the function $x^2-5$.

If we take $x~\ge-3$ we have to take the function 2x.

What is the range of a function?

The set of the outputs of a function is called the range of a function.

Example: f(x) = 3x + 1 where x = 1,2,3

Range of f(x) = {4, 7, 10}.

We see that the x values we can take for the given function are:

$x < -3\\x\geq-3$

We see that x can take values as:

-∞ ........ ,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, .......∞

We can say also write as x ∈ ( -∞, ∞ ).

Let's take x values of $x\geq-3$

For x = -3, g(x) = 2 x -3 = -6.

For x = -2, g(x) = 2 x -2 = -4.

For x = -1, g(x) = 2 x -1 = -2.

For x = 0, g(x) = 2 x 0 = 0

For x = 1, g(x) = 2 x 1 = 2.

For x = 2, g(x) = 2 x 2 = 4 and so on.

If we take $x \geq-3$ we see that g(x) will tend to  + ∞.

Lets take x values of x < -3.

For x = -4, g(x) = -4 x -4  - 5 = 16 - 5 = 11.

For x = -5, g(x) = -5x-5 -  5  = 25 - 5 = 20.

For x = -6, g(x) = -6 x -6 -  5 = 36 - 5 = 31.

For x = -7 g(x) = -7 x -7 - 5 = 49 - 5 = 44 and so on.

If we take x < -3 we see that g(x) tend to + ∞.

Thus for the given x values, the lowest value of g(x) is -6, and g(x) values keep on increasing.

Thus the range of a piecewise function g(x) for x < -3 and $x\geq-3$ is ( -6, ∞ ).      

Learn more about the range of a function here:

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