Question

Please help, algebra 2

Answer 1

`\( f(x) = 2 \cdot (3^x) - 4 \)` is an unbounded, increasing exponential function with no asymptote, covering all real numbers.

The function f(x) is an unbounded, strictly increasing exponential function with a no horizontal asymptote of the form
`\( f(x) = 2 \cdot (3^x) - 4 \)` . The range of the function is all real numbers and it is strictly increasing on its domain of all real numbers. The end behavior on the LEFT side is as
`x` , −∞ and the end behavior on the RIGHT side is as
`x`, +∞. (just fill in the blanks)

The function
`\( f(x) = 2 \cdot (3^x) - 4 \)` is an exponential function with a base of 3. Exponential functions of this form exhibit specific characteristics. First, there is no horizontal asymptote because, as
`\( x \)` approaches both positive and negative infinity, the function grows without bound. The exponential term
`\( 3^x \)` ensures unbounded growth.

The range of the function is all real numbers. As the exponential term covers the entire range of positive real numbers, the constant term
`\( -4 \)`allows the function to take on any real value. The function is strictly increasing across its domain of all real numbers due to the positive base of the exponential term (3).

On the left side, as
`\( x \)` approaches negative infinity, the function's end behavior is
`\( -\infty, -4 \)`, indicating unbounded downward growth approaching the constant term. On the right side, as
`\( x \)` approaches positive infinity, the end behavior is
`\( \infty, -4 \)`, signifying unbounded upward growth toward the constant term.

In summary, the function
`\( f(x) = 2 \cdot (3^x) - 4 \)` is an unbounded, strictly increasing exponential function with no horizontal asymptote and a range encompassing all real numbers.