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Exponential functions are mathematical expressions where a base number is raised to the power of a variable or exponent, commonly expressed in the form \( f(x) = a^x \). In the case of our exercise, we have the function \( f(x) = 2(3^x) - 18 \). Here, the base is 3, and the exponent is the variable \( x \).Understanding exponential functions is crucial because they model many real-world situations, such as population growth and radioactive decay, where changes happen at an exponential rate. The base value impacts how steep the curve of the graph is. For instance, a higher base results in a steeper curve. In our function, because the base is greater than 1, this means \( f(x) \) will increase as \( x \) increases.Exponential functions have some unique properties:

• They are always positive if the base is a positive number and the exponent is real.
• The graph features a horizontal asymptote, meaning it approaches a certain value but never actually reaches it; in many cases, this value is zero as \( x \) approaches negative infinity.
• The function grows rapidly, which is seen as the value of \( x \) increases. This is evident in the rapid rise of \( 3^x \).

Reviewing these properties helps make it clear why solving for \( x \) with an exponential function typically involves using logarithms or recognizing patterns in powers, as we saw when \( 3^x = 9 \) is simplified to \( 3^2 \).

Inequalities are expressions involving relationships of less than, greater than, or equal to. In our exercise, the task was to solve for \( f(x) < 0 \) and \( f(x) \geq 0 \) when \( f(x) = 2(3^x) - 18 \).When solving inequalities involving exponential functions, one core strategy is to first solve the equality \( f(x) = 0 \) to find where the function crosses the x-axis. In our function, we found that \( x = 2 \) when \( f(x) = 0 \). This critical point is essential because it divides the number line into segments that need to be considered separately: typically regions greater and less than this point.To find \( f(x) < 0 \) or \( f(x) \geq 0 \), consider the following:

• For \( f(x) < 0 \), you look for sections of the graph where the function dips below the x-axis. In our case, for \( x < 2 \), the graph is below the x-axis, indicating \( f(x) < 0 \).
• For \( f(x) \geq 0 \), you look for sections where the function is atop or intersects the x-axis. Here, \( x \geq 2 \) shows that the graph is above or touching the x-axis, indicating \( f(x) \geq 0 \).

Graphical translation of inequalities allows visualizing where these conditions are met easier than algebraic manipulation alone, aiding in ensuring accuracy.

Graphical analysis is a powerful tool for solving equations and inequalities, particularly when dealing with exponential functions like \( f(x) = 2(3^x) - 18 \). It involves sketching the graph of a function and using this visual representation to identify key properties such as where the function is positive, negative, or zero.When you plot \( y = f(x) \), begin by finding points that will shape your graph, such as intercepts and asymptotes. In our function, the x-intercept occurs at \( x = 2 \) because that is the value when \( f(x) = 0 \). For exponential functions, it's also useful to consider what happens as \( x \) approaches positive and negative infinity:

• As \( x \rightarrow -\infty \), \( 3^x \rightarrow 0 \), so \( f(x) \rightarrow -18 \).
• As \( x \rightarrow \infty \), \( 3^x \) rapidly increases, and so does \( f(x) \).

Graphically solving equations like \( f(x) = 0 \) and inequalities like \( f(x) < 0 \) becomes more intuitive. From the graph, you clearly see where \( f(x) \) crosses the x-axis and where it lies below or above it. This approach provides a straightforward bridge between algebraic calculations and real-world contexts. Visualization confirms algebraic solutions and reinforces understanding of function behavior across different values of \( x \).