Question

(a) If \(f(x)=5^{x-1}\) and \(g(x)=2^{x+1},\) graph \(f\) and \(g\) on the same Cartesian plane. (b) Find the point(s) of intersection of the graphs of \(f\) and \(g\) by solving \(f(x)=g(x) .\) Label any intersection points on the graph drawn in part (a). (c) Based on the graph, solve \(f(x)>g(x)\).

Short answer

Intersection point: \((1.7095, 2.836)\). For \(f(x) > g(x)\), \(x < 1.7095\).

Step-by-step solution

- Define the functions

The given functions are \(f(x) = 5^{x-1}\) and \(g(x) = 2^{x+1}\).

- Graph the functions separately

Plot \(f(x) = 5^{x-1}\) and \(g(x) = 2^{x+1}\) on the Cartesian plane. Note that as \(x\) increases, both functions grow exponentially. Evaluate a few points for each function to plot them accurately, for example, \(f(x)\) at \(x = -1, 0, 1, 2\) and so on, and similarly for \(g(x)\).

- Equate the functions to find points of intersection

To find the points of intersection, solve the equation: \[5^{x-1} = 2^{x+1}\].

- Solve for x

Take the logarithm of both sides (base 10 or base e): \[\begin{aligned}\log (5^{x-1}) &= \log (2^{x+1})\ (x-1)\cdot \log 5 &= (x+1)\cdot \log 2\ x\cdot \log 5 - \log 5 &= x\cdot \log 2 + \log 2\ x\cdot \log 5 - x\cdot \log 2 &= \log 2 + \log 5\ x(\log 5 - \log 2) &= \log 2 + \log 5 \ x &= \frac{\log 2 + \log 5}{\log 5 - \log 2} \ \end{aligned}\].

- Calculate x

Calculate the value of \(x\): \[x \approx \frac{\log 2 + \log 5}{\log 5 - \log 2} \approx 1.7095\].

- Determine the intersection point

Substitute \(x\) back into one of the original functions to find the y-coordinate. Using \(f(x)\): \[f(1.7095) \approx 5^{1.7095-1} = 5^{0.7095} \approx 2^{2.7095 - 1} \approx 2.836\]. Therefore, the intersection point is approximately \((1.7095, 2.836)\).

- Label the points on the graph

Label the intersection point \((1.7095, 2.836)\) on the graphs of \(f\) and \(g\).

- Analyze the graph for \(f(x) > g(x)\)

Based on the graph, determine where the function \(f(x)\) is above \(g(x)\). Since \(f(x)\) intersects \(g(x)\) at \(x \approx 1.7095\), for \(f(x) > g(x)\), \(x\textless1.7095\).

Key concepts

Function Graphing

Understanding how to plot functions is essential for visualizing mathematical concepts. Here, we deal with two exponential functions: \( f(x) = 5^{x-1} \) and \( g(x) = 2^{x+1} \). To graph these, start by evaluating each function at several key points.
For example:

• For \( f(x) \) at \( x = -1, 0, 1, 2 \), calculate \( f(-1) = 5^{-2} = 0.04 \), \( f(0) = 5^{-1} = 0.2 \), \( f(1) = 5^{0} = 1 \), \( f(2) = 5^{1} = 5 \).
• For \( g(x) \): calculate \( g(-1) = 2^{0} = 1 \), \( g(0) = 2^{1} = 2 \), \( g(1) = 2^{2} = 4 \), \( g(2) = 2^{3} = 8 \).

Then, plot these points on a Cartesian plane and draw smooth curves through them. Label each function clearly on the graph.
This will help you see how each function behaves and where they might intersect.

Intersection of Functions

The point(s) where two functions intersect is crucial in many problems. It often represents where they have the same value.
To find the intersection of \( f(x) = 5^{x-1} \) and \( g(x) = 2^{x+1} \), solve the equation \( 5^{x-1} = 2^{x+1} \).
Taking logarithms of both sides helps: \[ \log(5^{x-1}) = \log(2^{x+1}) \] This simplifies to: \[ (x-1) \cdot \log 5 = (x+1) \cdot \log 2 \] Solving for \( x \), you get: \[ x = \frac{\log 2 + \log 5}{\log 5 - \log 2} \] Calculating this gives \( x \approx 1.7095 \).
Substitute this back into one function to find \( y \). Using \( f(x) \): \[ f(1.7095) \approx 5^{1.7095 - 1} = 2.836 \] Thus, the intersection point is approximately \( (1.7095, 2.836) \).
Marking this on your graph provides a clear visual understanding of where the functions meet.

Solving Exponential Equations

Solving exponential equations often requires logarithms. Logarithms transform products, quotients, and powers into simpler terms, making the equation solvable.
For example, after equating \( 5^{x-1} = 2^{x+1} \), taking logarithms of both sides transforms the equation into: \[ \log(5^{x-1}) = \log(2^{x+1}) \] Applying the power rule: \[ (x-1) \cdot \log 5 = (x+1) \cdot \log 2 \] Isolate \( x \): \[ x \cdot \log 5 - \log 5 = x \cdot \log 2 + \log 2 \] Combine like terms: \[ x(\log 5 - \log 2) = \log 2 + \log 5 \] Solve for \( x \): \[ x = \frac{\log 2 + \log 5}{\log 5 - \log 2} \] Plugging the values of logs into a calculator gives \( x \approx 1.7095 \).
This value can then be used to find corresponding \( y \) values in the functions.

Inequalities in Graphs

Determining where one function is greater than another involves analyzing their graphs.
For \( f(x) > g(x) \), look at their intersection point. Here, \( f(x) = 5^{x-1} \) intersects \( g(x) = 2^{x+1} \) at \( x \approx 1.7095 \).
By examining the graph, we can see that:

• For \( x \textless 1.7095 \), the values of \( f(x) \) are above \( g(x) \).
• For \( x > 1.7095 \), the values of \( f(x) \) are below \( g(x) \).

Thus, the solution to \( f(x) > g(x) \) is \( x < 1.7095 \).
Simply put, before the intersection point, \( f \) grows faster and stays above \( g \).