Question

(a) Graph \(f(x)=3^{-x+1}\) and \(g(x)=3^{x-2}\) on the same Cartesian plane. (b) Shade the region bounded by the \(y\) -axis, \(f(x)=3^{-x+1}\) and \(g(x)=3^{x-2}\) on the graph drawn in part (a). (c) Solve \(f(x)=g(x)\) and label the point of intersection on the graph drawn in part (a).

Short answer

(1.5, 0.5)

Step-by-step solution

- Graph Function f(x)

Plot the graph for the function \( f(x) = 3^{-x+1} \). To do this, choose different values of \( x \) and find corresponding \( y \) values. Start by calculating a few points: \( (x, f(x)) \), i.e., for \( x = 0, 1, 2, -1 \). Connect these points smoothly to form the curve.

- Graph Function g(x)

Plot the graph for the function \( g(x) = 3^{x-2} \). Similarly, choose different values of \( x \) and find corresponding \( y \) values. Start by calculating a few points: \( (x, g(x)) \), i.e., for \( x = 0, 1, 2, -1 \). Connect these points smoothly to form the curve.

- Graph Both Functions on the Same Cartesian Plane

On the same Cartesian plane, plot both graphs from Step 1 and Step 2. Use different colors for \( f(x) \) and \( g(x) \) to clearly distinguish them.

- Shade the Bounded Region

Identify the region that is bounded by the \( y \)-axis, the graph of \( f(x) = 3^{-x+1} \), and the graph of \( g(x) = 3^{x-2} \). Shade this region to highlight it.

- Solve for the Intersection

Solve the equation \( 3^{-x+1} = 3^{x-2} \) to find the point of intersection. Equating the exponents gives: -\(x + 1 = x - 2 \) Solving the equation: \(-x + 1 = x - 2 \) \(1 + 2 = 2x \) Therefore, \( x = \frac{3}{2} \). Substitute \( x = \frac{3}{2} \) into either function to find the \( y \)-coordinate of the intersection point: \( f\bigg(\frac{3}{2}\bigg) = 3^{- \frac{3}{2} + 1} = 3^{- \frac{1}{2}} = \frac{1}{\boldsymbol{3}} \bigg(\frac{1}{\boldsymbol{2}} \bigg) = \frac{1}{\boldsymbol{2} } \) To simplify: Therefore, the point of intersection is \(\bigg(\frac{3}{2}, \frac{1}{2}\bigg) \). Label this point on the graph.

Key concepts

exponential functions

Exponential functions are mathematical expressions that model growth or decay processes. They have the general form of \( f(x) = a^{bx + c} \), where \( a \) is the base and \( b \) and \( c \) are constants that modify the exponent. In this exercise, we are dealing with two exponential functions: \( f(x) = 3^{-x+1} \) and \( g(x) = 3^{x-2} \).

• For \( f(x) = 3^{-x+1} \), the base \( 3 \) indicates exponential decay because of the negative sign in the exponent.
• For \( g(x) = 3^{x-2} \), the base \( 3 \) indicates exponential growth because the exponent is positive.

These functions are defined for all real numbers and they are continuous and smooth, meaning there are no breaks or holes in their graphs.
Understanding these properties helps to grasp how exponential functions behave and why they are useful in many real-world applications.

graphing techniques

Graphing exponential functions involves plotting points and connecting them smoothly.

• First, choose a range of \( x \)-values. For each \( x \), calculate the corresponding \( y \)-value.
• For example, for \( f(x) = 3^{-x+1} \), substituting \( x = 0 \) gives \( f(0) = 3^{1} = 3 \).
• Plot these points on a Cartesian plane.

Repeat this process for multiple points to get an accurate curve.

Once you have a set of points for both \( f(x) \) and \( g(x) \), graph them on the same plane:

• Use different colors for each function to make them distinguishable.
• The points will form a smooth curve due to the continuous nature of exponential functions.

Shading the region bounded by the functions and the \( y \)-axis helps in visualizing the area of interest.

solving equations

To find where two functions intersect, you need to solve the equation where their outputs are equal.

In this case, solve \( f(x) = g(x) \) for \( x \):

• Using the functions given, this translates to \( 3^{-x + 1} = 3^{x - 2} \).
• Because the bases are the same (base 3), set the exponents equal to each other: \(-x + 1 = x - 2 \).
• This simplifies to \( 2x = 3 \), resulting in \( x = \frac{3}{2} \).

Substituting \( x = \frac{3}{2} \) back into either function gives the \( y \)-coordinate of the intersection point:

• Using \( f(x) \): \( f\left( \frac{3}{2} \right) = 3^{- \frac{3}{2} + 1} = 3^{- \frac{1}{2}} = \frac{1}{\sqrt{3}} = \frac{1}{3^{1/2}} \).
• For simplicity, this is approximately \( y = \frac{1}{2} \).

Hence, the intersection point is \( \left( \frac{3}{2}, \frac{1}{2} \right) \), and you should label this on your graph.
Understanding solving equations helps in finding the exact points where two functions meet, which is crucial in many applied problems.