How do you find the domain and range of #root5(-4-7x)#?

Question

Nathan Axel · Accepted Answer

See below

Since the radicand can have negative values because this is an odd number root, the domain is: #{x in RR }# or #(-oo,oo)# For the range we observe what happens as x goes to #+-oo# as: #x->oo# , #color(white)(8888)-4-7x->-oo# as: #x->-oo# , #color(white)(8888)-4-7x->oo# Consequently, the range is: #{f(x) in RR}# or #(-oo,oo)# The graph of #f(x)=root(5)(-4-7x) # confirms this: y=root(5)(-4-7x) [-10, 10, -5, 5]}

Nathan Axel · Answer

undefined To find the domain and range of the function $ \sqrt{5(-4-7x)} $, follow these steps:

1. **Domain**: Determine the values of $ x $ for which the expression under the square root is valid. In this case, $ -4 - 7x $ must be non-negative since the square root of a negative number is undefined in the real number system.

$$ -4 - 7x \geq 0 $$

Solve the inequality for $ x $ to find the domain.

2. **Range**: For the range, consider the possible outputs of the function. Since the square root of a non-negative number is always non-negative, the range of the function will be non-negative real numbers.

$$ \sqrt{5(-4-7x)} \geq 0 $$

Solve this inequality to find the range.

Perform the calculations to determine the domain and range accordingly.