1. The function is given as $f(x)=3^{x}$. Determine the functions $g(x)=f(x+1)$ and $h(x)=4 f(x)-3$ and calculate the intersection point of their graphs.
1. Since $g(x)=f(x+1)$, we get $g(x)$ by substituting $x+1$ for $x$ in $f(x)$. This gives us $g(x)=3^{x+1}$. Similarly, we get $h(x)=4 \cdot 3^{x}-3$. To find the intersection points of the graphs of the functions $g(x)$ and $h(x)$, we set the equations of both functions equal to each other. This gives us the equation $3^{x+1}=4 \cdot 3^{x}-3$. We simplify the equation and get the solution $x=1$. We then calculate $y$. We get the intersection point $P(1,9)$.
Writing the function $g(x)=3^{x+1}$ ..... 1 point
Writing the function $h(x)=4 \cdot 3^{x}-3$ ..... 1 point
Setting up the equation $3^{x+1}=4 \cdot 3^{x}-3$ ..... 1 point
Simplifying the equation $3=3^{x}$ ..... 1 point
Solution: $x=1$ ..... 1 point
Writing the intersection point $P(1,9)$ ..... 1 point