Questions: h(x)=ln(4x+17)-5

Solution Steps

To determine the domain of the function \( h(x) = \ln(4x + 17) - 5 \), we need to find the values of \( x \) for which the expression inside the logarithm is positive, i.e., \( 4x + 17 > 0 \). For the range, since the logarithm function can take any real number as its output and the subtraction of 5 shifts it vertically, the range of \( h(x) \) is all real numbers.

To find the domain of the function \( h(x) = \ln(4x + 17) - 5 \), we need to ensure that the argument of the logarithm is positive:

\[ 4x + 17 > 0 \]

Solving this inequality:

\[ 4x > -17 \implies x > -\frac{17}{4} \]

Thus, the domain of \( h(x) \) is:

\[ \left(-\frac{17}{4}, \infty\right) \]

The function \( h(x) \) includes a logarithmic component, which can take any real number as its output. Since the logarithm can approach negative infinity and the subtraction of 5 only shifts the output downwards, the range of \( h(x) \) remains all real numbers:

\[ \text{Range} = \mathbb{R} \]

Final Answer