Question

question 2b (workbook). determine how many real number solutions each $f(x) = 0$, $g(x) = 0$, and $h(x) = 0$ has. explain how you know. $f(x) = 0$ has elija su respuesta... solution(s) because the graph has elija su respuesta... x-intercept(s) $g(x) = 0$ has elija su respuesta... solution(s) because the graph has elija su respuesta... x-intercept(s) $h(x) = 0$ has elija su respuesta... solution(s) because the graph has elija su respuesta... x-intercept(s) (dropdown options for $f(x)=0$ has: 0, 2, 1, -2, -1)

Explanation:

Step1: Recall the relationship between roots of \( f(x) = 0 \) and x - intercepts

The real - number solutions of the equation \( y = f(x)\) (i.e., \( f(x)=0\)) are the x - coordinates of the x - intercepts of the graph of \( y = f(x)\). In other words, the number of real - number solutions of \( f(x)=0\) is equal to the number of x - intercepts of the graph of \( y = f(x)\). The same logic applies to \( g(x)=0\) and \( h(x)=0\).

Step2: Analyze the number of solutions based on x - intercepts

• For \( f(x)=0\): If we assume the graph of \( y = f(x)\) has, say, \( n\) x - intercepts, then \( f(x)=0\) has \( n\) real - number solutions. For example, if the graph of \( f(x)\) has 0 x - intercepts, then \( f(x)=0\) has 0 solutions; if it has 1 x - intercept, \( f(x)=0\) has 1 solution; if it has 2 x - intercepts, \( f(x)=0\) has 2 solutions.
• For \( g(x)=0\): Similarly, the number of real - number solutions of \( g(x)=0\) is equal to the number of x - intercepts of the graph of \( y = g(x)\).
• For \( h(x)=0\): The number of real - number solutions of \( h(x)=0\) is equal to the number of x - intercepts of the graph of \( y = h(x)\).

Since the problem is about finding the number of real - number solutions of polynomial (or function) equations by looking at the number of x - intercepts of their graphs, we use the concept that the x - intercepts of the graph of \( y = f(x)\) correspond to the real roots of \( f(x)=0\).

Answer:

To determine the number of real - number solutions for \( f(x)=0\), \( g(x)=0\), and \( h(x)=0\):

• For \( f(x)=0\): The number of real - number solutions is equal to the number of x - intercepts of the graph of \( y = f(x)\). For example, if the graph of \( f(x)\) has \( n\) x - intercepts, \( f(x)=0\) has \( n\) real solutions.
• For \( g(x)=0\): The number of real - number solutions is equal to the number of x - intercepts of the graph of \( y = g(x)\).
• For \( h(x)=0\): The number of real - number solutions is equal to the number of x - intercepts of the graph of \( y = h(x)\).

(Note: Since the actual graph is not shown, but the general principle is that the number of real solutions of \( y = k(x)\) (where \( k\) is \( f\), \( g\), or \( h\)) is equal to the number of x - intercepts of the graph of \( y=k(x)\). For example, if the graph of \( f(x)\) has 0 x - intercepts, \( f(x)=0\) has 0 solutions; if 1 x - intercept, 1 solution; if 2 x - intercepts, 2 solutions, etc. The same for \( g(x)\) and \( h(x)\).)