A rational function is a function that contains at least one rational expression, such as If the dependent variable of the function is exchanged for a constant, say the result is a rational equation,which can be solved graphically. This is done by first graphing the function then finding the -coordinate(s) of the point(s) on the graph that has the -coordinate Then, the -coordinate(s) is the solution to the equation.
Solve the rational equation graphically.
When solving an equation graphically, we first rearrange the equation to get all variable-terms on one side and all constants on the other side. The left-hand side of the equation can now be expressed as a function, which we graph in a coordinate plane.
The solution to the equation is the -value of the point where the function equals We'll mark this point on the graph.
To begin, it's necessary to get the variable terms out of the denominator(s). Here, is in the denominator of three different ratios. To get out of the denominator of the first ratio, multiply the equation by This yields Notice that the variable term is no longer stuck in the denominator of the first ratio. The other denominators and must be handled in the same way. This leads to multiplying the entire equation by and which can be done in one step. Consider multiplying the first ratio by all three denominators. As can be seen, the denominator of the ratio is eliminated and what remains is the product of the numerator and the other denominators. This will happen for each ratio in the equation. Now that the work has been explained, solving the equation can begin.
Now that the variable terms have been consolidated, the equation can be solved. Notice here that the resulting equation is quadratic. This will not always be the case. The type of equation determines which method to use. Here, square roots will be used.
Tamara and Jamie are trimming the hedges in their front yard this weekend. Working together, this task takes hours. If Tamara works alone, it will take her hours. How long would it take Jamie to trim the hedges on her own?
Determine if is invertible. Then, find the inverse of
We can move an imaginary horizontal line across the graph. If it intersects the graph at more than one point anywhere, the inverse is not a function. We'll draw an arbitrary horizontal line. Sometimes more than one must be drawn.