Question

which of the statements is true?
oa. as x approaches positive infinity, h(x) exceeds f(x) and g(x).
ob. as x approaches positive infinity, f(x) converges with g(x).
oc. as x approaches positive infinity, f(x) exceeds g(x) and h(x).
od. as x approaches positive infinity, g(x) exceeds f(x) and h(x).

Explanation:

Step1: Analyze the behavior of \(h(x)\)

Looking at the graph, \(h(x)\) is a linear function with a negative slope. As \(x\to+\infty\), \(h(x)\to-\infty\).

Step2: Analyze the behavior of \(f(x)\)

\(f(x)\) is an exponential - like function (increasing but at a slower rate compared to a quadratic). As \(x\to+\infty\), \(f(x)\) increases, but its growth rate is less than that of a quadratic function.

Step3: Analyze the behavior of \(g(x)\)

\(g(x)\) is a quadratic function \(y = ax^{2}+bx + c\) (in this case, \(a>0\)). For a quadratic function \(y = ax^{2}+bx + c\) with \(a>0\), as \(x\to+\infty\), \(y = g(x)\to+\infty\) and \(g(x)\) grows faster than a linear function (since the degree of a quadratic function \(n = 2\) and the degree of a linear function \(m=1\) and for large \(x\), \(ax^{n}\) dominates \(bx^{m}\) when \(n>m\)) and also faster than an exponential - like function (in the long - run, for \(y = ax^{2}\) and \(y = b\cdot k^{x}\) with \(a,b,k>0,k
eq1\), the quadratic will be overtaken by the exponential for very large \(x\) if \(k > 1\), but in the context of common basic functions, among linear \(y=mx + b\), quadratic \(y = ax^{2}+bx + c(a>0)\) and a non - linear increasing function \(y = f(x)\) (not exponential here), the quadratic will out - grow the linear and the non - linear (non - exponential) function).

Since \(h(x)\to-\infty\) as \(x\to+\infty\), and \(g(x)\) grows faster than \(f(x)\) as \(x\to+\infty\) (quadratic growth rate is higher than the non - exponential non - quadratic growth rate in the graph shown).

Answer:

D. As \(x\) approaches positive infinity, \(g(x)\) exceeds \(f(x)\) and \(h(x)\)