We were doing review in class today and I remembered this one question I never got(though I don't have the specific question with me anymore).
1. It was a composite function question that asked for 'a' when f(g(a))=1. We were provided with the graphs of f(x) and g(x) and there were no equations for either one of those two functions, and the two graphs were reciprocal functions with some type of transformation(really hard to decipher)
So how could I approach this question? Algebraically? We are allowed graphing calculators, but I don't know how to use one.
2. There was a graph with zeros of 5 and -1, and it asked least amount of units it needed to shift horizontally so both of the zeros are positive.
Is 0 a positive number? If so, then I could shift it 1 to the right and get 6 and 0 as the new x intercepts instead.
1. To find 'a' when f(g(a)) = 1, you need to determine the value of 'a' that makes the composition of the two functions equal to 1.
Here's an algebraic approach to solving this type of question:
Step 1: Identify the values of 'a' that correspond to points where g(a) equals 1. This means we are looking for values of 'a' where g(a) intersects the line y = 1 on the graph.
Step 2: Once you have identified the values of 'a' from Step 1, substitute them into f(g(a)) and check if it equals 1. This will give you the specific value of 'a' that satisfies the equation.
If you have the graphs of f(x) and g(x) but not the equations, you can use the graphical approach to estimate the values of 'a'. Use the intersection points between g(a) and the line y = 1 to determine the x-coordinates of those points.
2. In the context of positive and negative numbers, 0 is considered to be neither positive nor negative. Therefore, to make both zeros (5 and -1) positive, you would need to shift the graph to the right. Shifting the graph 1 unit to the right would indeed result in new x-intercepts of 6 and 0.
To approach the first question regarding composite functions with graphs, you have a few options. One way is to use algebraic manipulation and the properties of composite functions. Here's a step-by-step approach:
1. Start by identifying the values of 'a' that make g(a) equal to a specific x-value on the graph of g(x) that you were given. Look for the x-values that correspond to the point(s) on the graph where g(x) crosses that x-value.
2. Once you have found the x-value(s) on the graph of g(x) that corresponds to g(a), substitute it into the function f(x). This will give you f(g(a)).
3. Set f(g(a)) equal to 1 and solve for 'a'.
If the graphs of f(x) and g(x) were difficult to decipher or understand, using a graphing calculator could be helpful. Here's how you can use a graphing calculator to find 'a' when f(g(a))=1:
1. Input the equation or function for g(x) into your graphing calculator. Make sure you understand how to enter equations or functions into your specific calculator.
2. Use the calculator to graph g(x) and identify the x-values at which the graph intersects your desired x-value.
3. Once you have found the x-values for g(a), input the equation or function for f(x) into your calculator.
4. Now, compose the two functions by entering f(g(x)) into the calculator and use it to graph the composite function.
5. Look for the y-value of 1 on the composite graph and read off the corresponding x-value. This will give you the value of 'a' you are looking for.
For the second question about shifting a graph horizontally, it's important to clarify the definition of positive numbers. In mathematics, positive numbers are greater than zero. Therefore, zero (0) is not considered positive.
To shift the graph so that both zeros are positive, you would need to move it left, not right. In your case, shifting it one unit to the left would result in the x-intercepts at 4 and -2, both of which are positive numbers.