# let f(x)=9x-2 and g(x)=√3x+7 find the domain of f and g 2.evaluate f(-2) f -1/3)

## Why did the chicken go to the domain party? Because it wanted to find out where the fun was happening!

The domain of a function is the set of all possible values for the variable. Let's find the domain of the functions f(x) and g(x).

For f(x) = 9x - 2, there are no restrictions on the possible values of x. In other words, any real number can be plugged into the function. So the domain of f(x) is all real numbers, or (-∞, ∞).

Now, for g(x) = √3x + 7, we need to be careful because square roots are only defined for non-negative numbers. To find the domain, we set the expression inside the square root, 3x + 7, greater than or equal to zero:

3x + 7 ≥ 0

Solving this inequality, we subtract 7 from both sides:

3x ≥ -7

Then, dividing by 3, we get:

x ≥ -7/3

So the domain of g(x) is all real numbers greater than or equal to -7/3, or [-7/3, ∞).

Now, let's evaluate f(-2) and f(-1/3) for some fun:

Plugging in x = -2 into f(x):

f(-2) = 9(-2) - 2

= -18 - 2

= -20

So, f(-2) = -20.

And plugging in x = -1/3 into f(x):

f(-1/3) = 9(-1/3) - 2

= -3 - 2

= -5

Hence, f(-1/3) = -5.

## To find the domain of a function, we need to determine the values of x for which the function is defined.

For the function f(x) = 9x - 2, it is a linear function, and linear functions are defined for all real values of x. Therefore, the domain of f(x) is all real numbers.

For the function g(x) = √(3x + 7), there is a square root involved. In order for the function to be defined, the expression inside the square root (√) must be non-negative (≥ 0). So we set up the inequality:

3x + 7 ≥ 0

Solving this inequality, we get:

3x ≥ -7

x ≥ -7/3

Therefore, the domain of g(x) is x ≥ -7/3.

Now let's evaluate f(-2) and f(-1/3) using their respective functions:

1. To evaluate f(-2), substitute -2 in place of x in the function f(x):

f(-2) = 9(-2) - 2

= -18 - 2

= -20

So, f(-2) equals -20.

2. To evaluate f (-1/3), substitute (-1/3) in place of x in the function f(x):

f(-1/3) = 9(-1/3) - 2

= -3 - 2

= -5

So, f(-1/3) equals -5.

## f is any old x BUT g will not work for x <0

9(-2) - 2 = ?

etc

## assuming the usual carelessness with parentheses, and thus that g(x) = √(3x+7)

since the domain of √x is x≥0,

the domain of g is 3x+7 ≥ 0, or x ≥ -7/3

the domain of f, as with all polynomials, is all real numbers.