Question

select the correct answer.
which function is increasing on the interval $(-\infty, \infty)$?
\\(\bigcirc\\) a. \\( j(x) = x^2 + 8x + 1 \\)
\\(\bigcirc\\) b. \\( f(x) = -3x + 7 \\)
\\(\bigcirc\\) c. \\( h(x) = 2^x - 1 \\)
\\(\bigcirc\\) d. \\( g(x) = -4(2^x) \\)

Explanation:

Step1: Analyze Option A

The function \( j(x)=x^{2}+8x + 1\) is a quadratic function with \(a = 1>0\). The vertex of the parabola is at \(x=-\frac{b}{2a}=-\frac{8}{2\times1}=- 4\). The function is decreasing on \((-\infty,-4)\) and increasing on \((-4,\infty)\), so it is not increasing on \((-\infty,\infty)\).

Step2: Analyze Option B

The function \(f(x)=-3x + 7\) is a linear function with slope \(m=-3<0\). A linear function with negative slope is decreasing on \((-\infty,\infty)\), so it is not increasing on \((-\infty,\infty)\).

Step3: Analyze Option C

The function \(h(x)=2^{x}-1\) is an exponential function with base \(a = 2>1\). For an exponential function \(y = a^{x}\) with \(a>1\), the function is increasing on \((-\infty,\infty)\). The function \(h(x)=2^{x}-1\) is a vertical shift of \(y = 2^{x}\) (shifted down by 1 unit), and the vertical shift does not change the increasing/decreasing nature. So \(h(x)\) is increasing on \((-\infty,\infty)\).

Step4: Analyze Option D

The function \(g(x)=-4(2^{x})\) is an exponential function. The base \(a = 2>1\), but there is a negative coefficient \(-4\). The function \(y = 2^{x}\) is increasing, and multiplying by a negative number reflects it over the \(x\)-axis, so \(g(x)=-4(2^{x})\) is decreasing on \((-\infty,\infty)\).

Answer:

C. \( h(x) = 2^{x} - 1\)