Question

In Exercises \(9-12,\) use a graph to find the zeros of the function. $$f(x)=3-2^{x}$$

Short answer

The zeros of the function can only be obtained graphically or using numerical methods due to the nature of the function.

Step-by-step solution

Graph the function

Plot the function \(f(x)=3-2^{x}\) on a graph. This function is a transformed exponential function.

Identify the zeros of the function

Look for the points where the function crosses the x-axis. These points are the zeros of the function.

Extract the x-coordinates of the zeros

Record the x-coordinates of the points where the function crosses the x-axis. These are the zeros of the function.

Key concepts

Graphing Exponential Functions

An understanding of how to graph exponential functions is essential for identifying their characteristics, such as the zeros. To start graphing an exponential function like \(f(x) = 3 - 2^{x}\), we first consider the basic form \(a^x\), where \(a\) is a positive constant. This function increases or decreases rapidly depending on whether \(a\) is greater than or less than 1, respectively.

The graph typically crosses the y-axis at \((0,1)\), but transformations can change this. For the given function, consider the horizontal shift and vertical displacement introduced by the subtraction of 2 to the power of \(x\) and the addition of 3. To graph it effectively, calculate a few key points by substituting values of \(x\) and then sketch the curve, observing how it approaches, but never reaches, the y-axis on the left and flattens out as it moves to the right.

Exponential Function Transformations

Translating Vertically and Horizontally

Exponential function transformations involve shifting, stretching, compressing, and reflecting the graph of the base function \(a^x\). With our example, \(f(x) = 3 - 2^{x}\), the '-2' affects the rate of increase or decrease of the function, while the addition of '3' vertically shifts the entire graph upwards by 3 units.

Observing Reflections and Stretching

When a negative is placed in front of the exponential component, as in our function, it reflects the graph over the y-axis, changing the direction of the curve. Additionally, if \(a\) were greater than 1, the function would stretch vertically, and if less than 1, it would compress.

X-Intercepts of Functions

The x-intercepts of a function are points where the graph of the function crosses the x-axis. These points represent the zeros of the function, which are the solutions to the equation when \(f(x) = 0\). In the context of our function \(f(x) = 3 - 2^{x}\), finding the x-intercept entails setting the function equal to zero and solving for \(x\).

By doing this, we are effectively solving the equation \(3 - 2^{x} = 0\). Adjusting the equation gives us \(2^{x} = 3\), which leads us to a power of 2 close to 3. Since 2 to the power of something slightly more than 1 is 3, we identify one zero of the function at approximately \(x = 1.58496\), which is the logarithm of 3 on the base of 2.

Algebraic Solving Techniques

To find the exact zeros of exponential functions algebraically, specific techniques can be employed. For the given function \(f(x) = 3 - 2^{x}\), where the aim is to find the value of \(x\) that makes \(f(x)\) equal to zero, we can isolate the exponential expression and use logarithms. Setting \(f(x)\) to zero gives us \(2^{x} = 3\).

Applying the logarithm of the base corresponding to the base of the exponent, in this case, base 2, allows us to solve for \(x\) as \(x = \log_2(3)\). This is an exact solution, which can then be approximated numerically if needed. Understanding the use of logarithms is crucial for these algebraic solving techniques, as they translate an exponential equation into a linear form that can be easily solved.