Question

x g(x)
-2 -2
-1 1
0 4
1 7
2 10
which statement correctly compares the graphs of the functions?
a. as the values of x increase, g(x) will eventually be greater than f(x).
b. the y-intercept of f(x) is greater than the y-intercept of g(x).
c. as the values of x increase, the rate of change of f(x) decreases and the rate of change of g(x) remains constant.
d. as the values of x increase, the rate of change of f(x) increases and the rate of change of g(x) remains constant.

Explanation:

Step1: Analyze g(x)'s rate of change

To find the rate of change of \( g(x) \), we use the formula for slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
For \( x=-2 \) to \( x = - 1 \): \( m=\frac{1-(-2)}{-1-(-2)}=\frac{3}{1} = 3 \)
For \( x=-1 \) to \( x = 0 \): \( m=\frac{4 - 1}{0-(-1)}=\frac{3}{1}=3 \)
For \( x = 0 \) to \( x=1 \): \( m=\frac{7 - 4}{1 - 0}=\frac{3}{1}=3 \)
For \( x = 1 \) to \( x = 2 \): \( m=\frac{10 - 7}{2 - 1}=\frac{3}{1}=3 \)
So the rate of change of \( g(x) \) is constant (\( m = 3 \)).

Step2: Analyze the nature of f(x) (assuming f(x) is a common function like exponential or quadratic, but since we know g(x) is linear with constant rate, we check the options)

• Option A: Without knowing f(x), we can't say. But since g(x) is linear and if f(x) is, say, linear with lower slope or non - linear, but we can't be sure. But let's check other options.
• Option B: We don't know the y - intercept of f(x), so we can't compare.
• Option C: If f(x) is, for example, an exponential function \( f(x)=a^x \) ( \( a>1 \) ) or a quadratic function \( f(x)=ax^{2}+bx + c \) ( \( a>0 \) ), the rate of change (slope of tangent or difference quotient) increases. But if f(x) is linear, its rate of change is constant. However, the key here is the rate of change of g(x) is constant (we proved it's 3) and for many common functions (like exponential, quadratic) the rate of change increases. But the option says "the rate of change of f(x) decreases" which is wrong.
• Option D: If f(x) is an exponential function (e.g., \( f(x)=2^x \)) or a quadratic function (e.g., \( f(x)=x^{2} \)), as \( x \) increases, the rate of change (for exponential, the derivative \( f^\prime(x)=2^x\ln(2) \) increases; for quadratic, the derivative \( f^\prime(x) = 2x \) increases) and the rate of change of \( g(x) \) is constant (we proved \( m = 3 \)). This option is correct.

Answer:

D. As the values of x increase, the rate of change of f(x) increases and the rate of change of g(x) remains constant.