Question

a linear function and an exponential function are shown. over which interval does the growth rate of the exponential function exceed the growth rate of the linear function? options: 1≤x≤2, x<1, 0≤x≤1, x>2

Explanation:

Step1: Find the growth rate of the linear function

The linear function passes through \((0,0)\) and \((2,4)\). The slope (growth rate) \(m\) of a linear function \(y = mx + b\) is calculated by \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Here, \(x_1 = 0,y_1 = 0,x_2=2,y_2 = 4\), so \(m=\frac{4 - 0}{2 - 0}=2\).

Step2: Find the growth rate of the exponential function

The exponential function passes through \((0,1)\) and \((1,2)\) and \((2,4)\). The general form of an exponential function is \(y = a\cdot b^x\). When \(x = 0\), \(y=1\), so \(a = 1\). When \(x = 1\), \(y = 2\), so \(2=1\cdot b^1\), then \(b = 2\). The exponential function is \(y = 2^x\). The growth rate (average rate of change) over an interval \([x_1,x_2]\) is \(\frac{y_2 - y_1}{x_2 - x_1}\).

• For interval \(x<1\) (take \(x = 0\) to \(x = 1\)): \(\frac{2 - 1}{1 - 0}=1\), which is less than 2.
• For interval \(0\leq x\leq1\): \(\frac{2 - 1}{1 - 0}=1\), less than 2.
• For interval \(1\leq x\leq2\): \(\frac{4 - 2}{2 - 1}=2\), equal to 2.
• For interval \(x>2\) (take \(x = 2\) to \(x = 3\), assume \(x = 3\), \(y=2^3 = 8\)): \(\frac{8 - 4}{3 - 2}=4\), which is greater than 2.

Answer:

\(x>2\)