Questions: Identify the Domain of the following functions. Write your answers in Interval Notation. Function g(x)=1/(x+1) f(x)=-3x^2+8 p(t)=-9t+9 h(x)=sqrt(x+2) Domain square square square square

Solution Steps

To find the domain of a function, we need to identify the set of all possible input values (x-values) for which the function is defined. For rational functions, the denominator cannot be zero. For square root functions, the expression inside the square root must be non-negative.

1. For \( g(x) = \frac{1}{x+1} \), the function is undefined when the denominator is zero, so \( x+1 \neq 0 \).

2. For \( f(x) = -3x^2 + 8 \), it is a polynomial function, which is defined for all real numbers.

3. For \( p(t) = -9t + 9 \), it is a linear function, which is also defined for all real numbers.

4. For \( h(x) = \sqrt{x+2} \), the expression inside the square root must be non-negative, so \( x+2 \geq 0 \).

The function \( g(x) = \frac{1}{x+1} \) is a rational function. The domain of a rational function is all real numbers except where the denominator is zero.

Set the denominator equal to zero and solve for \( x \):

\[ x + 1 = 0 \implies x = -1 \]

Thus, the domain of \( g(x) \) is all real numbers except \( x = -1 \).

In interval notation, the domain is:

\[ (-\infty, -1) \cup (-1, \infty) \]

The function \( f(x) = -3x^2 + 8 \) is a polynomial function. The domain of a polynomial function is all real numbers because polynomials are defined for every real number.

In interval notation, the domain is:

\[ (-\infty, \infty) \]

The function \( p(t) = -9t + 9 \) is a linear function. The domain of a linear function is all real numbers because linear functions are defined for every real number.

In interval notation, the domain is:

\[ (-\infty, \infty) \]

Final Answer