Problem 14

Explanation

We will analyze each statement one by one based on the functions $f(x) = 5 - x$ and $g(x) = 2^x - 1$.

Analysis of Statement 1

Statement: $g(x)$ is a linear line with a gradient of $2$.

The function $g(x) = 2^x - 1$ is an exponential function, not a linear function. A linear function has the general form $y = mx + c$, whereas this is an exponential form. Therefore, statement $(1)$ is incorrect.

Analysis of Statement 2

Statement: $f(x)$ and $g(x)$ intersect at $x > 0$.

To find the intersection point, we equate the two functions:

If we substitute $x = 2$:

Both functions have the same value of $3$ when $x = 2$. Since $2 > 0$, statement $(2)$ is correct.

Analysis of Statement 3

Statement: $f(x)$ is above $g(x)$ for all values of $x$.

Let's check a value $x > 2$, for example $x = 3$:

Here it is seen that $g(3) > f(3)$, which means the curve $g(x)$ is above $f(x)$. Therefore, the statement that $f(x)$ is always above $g(x)$ is incorrect. Statement $(3)$ is incorrect.

Analysis of Statement 4

Statement: The graphs of $f(x)$ and $g(x)$ intersect at $(2, 3)$.

As proven in the analysis of statement $(2)$, both graphs intersect when $x = 2$ and yield the value $y = 3$. Thus, the intersection point is $(2, 3)$. Statement $(4)$ is correct.

Conclusion

The correct statements are $(2)$ and $(4)$.