Question

select the correct answer from each drop-down menu. use the given functions represented by the equation and the graph to complete the statements. $f(x) = 2x^3 - 3x^2 + 5x + 7$ as $x$ approaches positive infinity, $f(x)$ approaches drop - down and $g(x)$ approaches drop - down. the $y$-intercept of function $f$ is drop - down the $y$-intercept of function $g$.

Explanation:

Step1: Analyze \( f(x) \) as \( x \to +\infty \)

The leading term of \( f(x) = 2x^3 - 3x^2 + 5x + 7 \) is \( 2x^3 \). For a cubic function with a positive leading coefficient (\( 2>0 \)), as \( x \to +\infty \), \( f(x) \to +\infty \).

Step2: Analyze \( g(x) \) as \( x \to +\infty \)

From the graph of \( g(x) \), as \( x \) approaches positive infinity, the graph goes downwards, so \( g(x) \to -\infty \).

Step3: Find y-intercept of \( f(x) \)

The y-intercept of a function is found by setting \( x = 0 \). For \( f(x) \), \( f(0)=2(0)^3 - 3(0)^2 + 5(0)+7 = 7 \).

Step4: Find y-intercept of \( g(x) \)

From the graph, when \( x = 0 \), the y - value (y - intercept) of \( g(x) \) is 8 (since the graph passes through \( (0,8) \)).

Step5: Compare y - intercepts

Since \( 7<8 \), the y - intercept of \( f(x) \) is less than the y - intercept of \( g(x) \).

Answer:

As \( x \) approaches positive infinity, \( f(x) \) approaches \( +\infty \) and \( g(x) \) approaches \( -\infty \). The \( y \)-intercept of function \( f \) is less than the \( y \)-intercept of function \( g \).