Solve this SAT question---75. The function $f$ is defined by $f(x) = 3^x$. The function $g$ is a increasing linear function. In the $xy$-plane, the graphs of $y = f(x)$ and $y = g(x)$ intersect at two points $(a, j)$ and $(b, k)$, where $j < k$. When $g(x) > f(x)$, which of the following must be true?
A) $x > k$
B) $x < j$
C) $x < j$ or $x > b$
D) $a < x < b$
视频信息
答案文本
视频字幕
Let's analyze this SAT problem step by step. We have an exponential function f of x equals 3 to the x, and an increasing linear function g of x. These two functions intersect at two points: a comma j and b comma k, where j is less than k. We need to determine when g of x is greater than f of x.
Now let's understand the behavior of these functions. The exponential function 3 to the x is convex and increasing, while g of x is a straight line. Since j is less than k and both functions are increasing, we know that a is less than b. The key insight is that between the two intersection points, the linear function lies above the exponential curve. This highlighted region shows where g of x is greater than f of x.
Now let's analyze each answer choice. Options A and B compare x to y-values j and k, which doesn't make sense. Option C mixes a y-value with an x-value inconsistently. Option D states that a is less than x which is less than b. This perfectly matches our analysis - g of x is greater than f of x exactly when x lies between the two intersection points a and b.
To summarize our solution: The exponential function f of x equals 3 to the x intersects the linear function g of x at two points. Between these intersection points a and b, the linear function lies above the exponential curve. Therefore, g of x is greater than f of x exactly when a is less than x which is less than b. The correct answer is D.