Question

paring functions
select all true statements based on the graph of the functions.
when ( x = 8 ), the value of ( g(x) ) exceeds the value of ( f(x) ).
as ( x ) increases, ( f(x) ) and ( g(x) ) both increase.
as ( x ) increases, ( g(x) ) will never exceed ( f(x) ).
a quantity increasing exponentially eventually exceeds a quantity increasing quadratically.
over the interval ( 4,5 ), the average rate of change of ( g(x) ) is greater than the average rate of change of ( f(x) )

Explanation:

Step1: Analyze the first statement

When \(x = 8\), observe the graph. The \(y -\)value (function value) of \(g(x)\) is higher than that of \(f(x)\). So, when \(x = 8\), \(g(x)>f(x)\).

Step2: Analyze the second statement

As \(x\) increases (moving from left - to - right on the \(x -\)axis), the \(y -\)values (function values) of both \(f(x)\) (a quadratic - like function, assuming \(f(x)=ax^{2}+bx + c,a>0\)) and \(g(x)\) (an exponential - like function, assuming \(g(x)=a\cdot b^{x},a>0,b > 1\)) are going up. So, as \(x\) increases, \(f(x)\) and \(g(x)\) both increase.

Step3: Analyze the third statement

Exponential functions \(y = a\cdot b^{x}(a>0,b>1)\) grow faster than quadratic functions \(y=ax^{2}+bx + c(a>0)\) as \(x\) gets large enough. So, the statement “As \(x\) increases, \(g(x)\) will never exceed \(f(x)\)” is false.

Step4: Analyze the fourth statement

The general form of an exponential function is \(y = a\cdot b^{x}(a>0,b > 1)\) and the general form of a quadratic function is \(y=ax^{2}+bx + c(a>0)\). Using the limit \(\lim_{x
ightarrow+\infty}\frac{ax^{2}+bx + c}{a\cdot b^{x}}=0\) (by L'Hopital's rule twice for the indeterminate form \(\frac{\infty}{\infty}\) if needed), a quantity increasing exponentially eventually exceeds a quantity increasing quadratically.

Step5: Analyze the fifth statement

The average rate of change of a function \(y = h(x)\) over the interval \([m,n]\) is \(\frac{h(n)-h(m)}{n - m}\).
For the interval \([4,5]\):
The average rate of change of \(f(x)\) is \(\frac{f(5)-f(4)}{5 - 4}=f(5)-f(4)\).
The average rate of change of \(g(x)\) is \(\frac{g(5)-g(4)}{5 - 4}=g(5)-g(4)\).
By looking at the vertical distances (since the denominator \(5 - 4=1\)) on the graph for \(x = 4\) and \(x = 5\), \(g(5)-g(4)>f(5)-f(4)\).

Answer:

When \(x = 8\), the value of \(g(x)\) exceeds the value of \(f(x)\); As \(x\) increases, \(f(x)\) and \(g(x)\) both increase; A quantity increasing exponentially eventually exceeds a quantity increasing quadratically; Over the interval \([4,5]\), the average rate of change of \(g(x)\) is greater than the average rate of change of \(f(x)\)