if a problem says f(x)=square root x^2-16; x less than or equal to -4.

Find domain and range.

to do this i set up the equation like x^2-16 less than or equal to -4. However by doing this I am runnning into complications findng domain and range. Am i doing it wrong can someone help me

f(x)=?( x^2-16)

let's first look at it

http://www.wolframalpha.com/input/?i=f(x)%3D%E2%88%9A(+x%5E2-16)

consider only the "blue" graph, which is the reals

clearly the domain is y ? 0

for the domain, x^2 - 16 ? 0
x^2 ? 16
x ? 4 OR x ? -4

If x ? -4 is part of the question, the use only the left part of the graph

To find the domain and range of a function, you need to analyze the restrictions on the input (domain) and the corresponding output values (range).

Given the function f(x) = √(x^2 - 16), where x ≤ -4, let's start by determining the domain.

For the square root function, the expression inside the square root (x^2 - 16) must be greater than or equal to zero because the square root of a negative number is undefined. So we set x^2 - 16 ≥ 0 and solve for x.

x^2 - 16 ≥ 0

To solve this inequality, we need to find the values of x that make the equation true.

First, express the inequality as two separate parts:

x^2 - 16 ≥ 0
(x - 4)(x + 4) ≥ 0

Now, we need to determine the intervals for which this inequality is true.

1. When (x - 4)(x + 4) > 0:

- If both factors are positive, the product is positive. Therefore, x > 4.

- If both factors are negative, the product is positive. However, it doesn't satisfy the given condition, x ≤ -4.

2. When (x - 4)(x + 4) = 0:

- If either factor is zero, the product becomes zero. So, x = -4 and x = 4.

- However, x = 4 is not within the given condition x ≤ -4.

Now, let's plot the values on a number line to determine the intervals that satisfy the inequality:

-------------------o---o-----------------------
-4 4

In summary, the domain for the given function is (-∞, -4] because it includes all real numbers less than or equal to -4.

To find the range, we consider the possible output values for f(x). Notice that the expression inside the square root, x^2 - 16, will always yield non-negative results for the given domain.

Since the square root function only gives non-negative values, the range of f(x) includes all non-negative real numbers. Therefore, the range is [0, ∞).

I hope this explanation helps in understanding how to find the domain and range of the given function.