Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=x-3\) and \(g(x)=x^{2}-6 x\). Find the \(x\) - and \(y\) -intercepts of the following. (a) \(f(x)\) (b) \(g(x)\) (c) \(f(x) g(x)\) (d) \(\frac{f(x)}{g(x)}\)

Step-by-Step Solution

Answer

The solutions are: For \(f(x)\), the x-intercept is 3 and y intercept is -3. For \(g(x)\), the x-intercepts are 0 and 6, and y intercept is 0. For \(f(x) g(x)\), the x-intercepts are 0, 3, and 6, and the y intercept is 0. For \(\frac{f(x)}{g(x)}\), the x-intercept is 3 and there's no y-intercept as the function is undefined at \(x=0\).

Step 1: Find the x- and y-intercepts for \(f(x)\)

For \(f(x)\), we set \(f(x) = 0\) to obtain the x-intercept, resulting in \(x - 3 = 0\). By solving this, we get \(x = 3\) which is the x-intercept. To get the y-intercept (value of \(f(x)\) when \(x = 0\)), we substitute \(x = 0\) into the equation \(f(x)\) to get \(f(0) = 0 - 3 = -3\), the y-intercept.

Step 2: Find the x- and y-intercepts for \(g(x)\)

For \(g(x)\), set \(g(x) = 0\) to find the x-intercepts. This gives us \(x^{2} - 6x = 0\). By factorizing \(x\) out we get \(x(x - 6) = 0\). This gives two solutions, \(x = 0\) and \(x = 6\) which are the x-intercepts. For the y-intercept, we substitute \(x = 0\) into \(g(x)\) to get \(g(0) = 0 - 0 = 0\).

Step 3: Find the x- and y-intercepts for \(f(x) g(x)\)

The function \(f(x) g(x) = (x - 3)(x^{2} - 6x)\). To find the x-intercepts, let \(f(x) g(x) = 0\) and solve for \(x\). This is the same as setting \(x - 3 = 0\) and \(x^{2} - 6x = 0\), hence \(x = 0, 3, 6\) are the x-intercepts. To find the y-intercept we put \(x = 0\) into \(f(x) g(x) = (0 - 3)(0 - 0)\) and compute the solution as \(0\).

Step 4: Find the x- and y-intercepts for \(\frac{f(x)}{g(x)}\)

The function \(\frac{f(x)}{g(x)} = \frac{x - 3}{x^{2} - 6x}\). To find the x-intercept, we set the numerator equal to zero, i.e., \(x - 3 = 0\). We get \(x = 3\) as the x-intercept. As this is a fraction, there is no y-intercept, as setting \(x = 0\) will result in an undefined number due to division by zero.

Key Concepts

Understanding Linear Functions

Linear functions, like the one given as \( f(x) = x - 3 \), are the simplest type of function since they graph as straight lines. The equation is in the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. Linear functions are defined by their constant rate of change, which means their slope is always the same, no matter where you are on the line.

To find the x-intercept of a linear function, you need to set the function equal to zero and solve for \( x \). This gives you the point where the line crosses the x-axis. For example, with \( f(x) = x - 3 \), setting \( x - 3 = 0 \) results in \( x = 3 \). This means the x-intercept is at 3.

For the y-intercept, substitute \( x = 0 \) into the function. For our function, \( f(0) = 0 - 3 = -3 \), so the y-intercept is -3.

Linear functions are straightforward to graph and analyze due to their predictable, unchanging slope.

Exploring Quadratic Functions

Quadratic functions like \( g(x) = x^2 - 6x \) form parabolas when graphed. These functions are characterized by their \( ax^2 + bx + c \) form. The presence of \( x^2 \) makes the equation a curve instead of a straight line.

The intercepts for quadratic functions include both possible x-intercepts (roots) and a y-intercept. To find the x-intercepts, set the function equal to zero and solve for \( x \). This quadratic function \( g(x) = x^2 - 6x \) can be factored as \( x(x - 6) = 0 \), so \( x = 0 \) and \( x = 6 \) are the intercepts.

The y-intercept is found by substituting \( x = 0 \), giving \( g(0) = 0 \).

• Quadratic equations can have up to two x-intercepts.
• The graph of these functions is a parabola.

Quadratics can either open upwards or downwards, depending on the sign of the \( ax^2 \) term.

Introduction to Rational Functions

Rational functions represent quotients of two polynomials. For instance, \( \frac{f(x)}{g(x)} = \frac{x - 3}{x^2 - 6x} \) is a rational function. These functions have features like asymptotes, which are lines that the graph approaches but never meets.

Finding intercepts for rational functions can be more involved. The x-intercept occurs when the numerator equals zero, so \( x - 3 = 0 \) leads us to \( x = 3 \). However, rational functions might not have a y-intercept. In this case, because the denominator becomes zero at \( x = 0 \), substituting into \( \frac{f(x)}{g(x)} \) results in an undefined expression, meaning there is no y-intercept.

Rational functions can produce more complex graphs due to their division and possibility of discontinuities. They often require more detailed analysis to fully understand their behavior.