Preview Activity 2.5.1

We will begin by considering a more familiar situation; namely, the function \(f(x) = x^2\text{,}\) which takes a real number \(x\) as an input and produces its square \(x^2\) as its output.

  1. What is the value of \(f(3)\text{?}\)

  2. Can we solve the equation \(f(x) = 4\text{?}\) If so, is the solution unique?

  3. Can we solve the equation \(f(x) = -10\text{?}\) If so, is the solution unique?

  4. Sketch a graph of the function \(f(x)=x^2\) in Figure 2.5.1

    Figure 2.5.1 Graph the function \(f(x)=x^2\) above.

  5. Remember that the range of a function is the set of all possible outputs. What is the range of the function \(f\text{?}\)

  6. We will now consider functions having the form \(g(x)=mx\text{.}\) Draw a graph of the function \(g(x) = 2x\) on the left in Figure 2.5.2.

    Figure 2.5.2 Graphs of the function \(g(x)=2x\) and \(h(x) = -\frac13 x\text{.}\)

  7. Draw a graph of the funcion \(h(x) = -\frac13 x\) on the right of Figure 2.5.2.

  8. Remember that composing two functions means we use the output from one function as the input into the other. That is, \(g\circ h(x) = g(h(x))\text{.}\) What function results from composing \(g\circ h(x)\text{?}\) How is the composite function related to the two functions \(g\) and \(h\text{?}\)

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