Question

The graph of a function \(f\)and \(g\) are given.

(a) State the values of \(f\left( { - 4} \right)\)and\(g\left( 3 \right)\).

(b) For what values of xis\(f\left( x \right) = g\left( x \right)\)?

(c) Estimate the solution of the equation\(f\left( x \right) = - 1\).

(d) On what interval \(f\)is decreasing?

(e) State the domain and range of\(f\).

(f) State the domain and range of\(g\).

Short answer

(a) The value of \(f\left( { - 4} \right)\) and \(g\left( 3 \right)\)are equal to \( - 2\)and\(4\).

(b) The functions \(f\left( x \right) = g\left( x \right)\) when \(x = - 2\)and\(x = 2\).

(c) Thevalues of \(x\)are\(\left( {0,\;3} \right)\).

(d) The function \(f\) is decreasing on the interval\(\left( {0,\;4} \right)\).

(e) The domain and range of \(f\)are\(\left( { - 4,\;4} \right)\), and\(\left( { - 2,\;3} \right)\), respectively.

(f) The domain and range of \(g\)are\(\left( { - 4,\;3} \right)\), and\(\left( {\frac{1}{2},\;4} \right)\), respectively.

Step-by-step solution

(a) Step 1: Given information

Consider the given graph.

State the value of \(f\left( { - 4} \right)\)and\(g\left( 3 \right)\).

The value of \(f\left( { - 4} \right)\) is equal to the value of the y-coordinate of the graph \(f\)at\(x = - 4\). From the graph, it can be observed that the value of \(y = - 2\)at\(x = - 4\). Therefore,\(f\left( { - 4} \right) = - 2\).

The value of \(g\left( 3 \right)\) is equal to the value of the y-coordinate of the graph \(g\)at\(x = 3\). From the graph, it can be observed that the value of \(y = 4\)at\(x = 3\). Therefore, \(g\left( 3 \right) = 4\).

Therefore, the value of \(f\left( { - 4} \right)\) and \(g\left( 3 \right)\)are equal to \( - 2\)and\(4\).

(b) Step 3: Find the value of x at which\(f\left( x \right) = g\left( x \right)\).

The functions \(f\left( x \right) = g\left( x \right)\) will be at the point of intersection. From the given graphs, it can be observed that the functions intersect at points \(\left( { - 2,\;1} \right)\)and\(\left( {2,\;2} \right)\).

Therefore, the functions \(f\left( x \right) = g\left( x \right)\) when \(x = - 2\)and\(x = 2\).

(c) Step 4: Find the solution to the equation\(f\left( x \right) =  - 1\).

For\(f\left( x \right) = - 1\), it is required to find the \(x\)-values on the graph where\(y = - 1\).

From the graph, it can be observed that \(y = - 1\) is at \(x = - 3\)and\(x = 4\).

Therefore, the solutions of the equation \(f\left( x \right) = - 1\) are \(x = - 3\)and\(x = 4\).

(d) Step 5: Find the interval at which f is decreasing

From the given graphs, it can be observed that as \(x\) increases from \(0\)to\(4\), the value of \(y\) is decreasing from \(3\) to\( - 1\). This means that the function \(f\) is decreasing on the interval\(\left( {0,\;4} \right)\).

Therefore, the function \(f\) is decreasing on the interval\(\left( {0,\;4} \right)\).

(e) Step 6: State the domain and range of\(f\).

From the graph, it can be observed that the function \(f\) is defined when\( - 4 \le x \le 4\). Therefore, the domain of the function \(f\)is\(\left( { - 4,\;4} \right)\).

The function takes all values from \( - 2\)to\(3\). Therefore, the range of the function \(f\)is\(\left( { - 2,\;3} \right)\).

Therefore, the domain and range of \(f\)are\(\left( { - 4,\;4} \right)\), and\(\left( { - 2,\;3} \right)\), respectively.

(f) Step 7: State the domain and range of\(g\).

From the graph, it can be observed that the function \(g\)exists for all the values of\( - 4 \le x \le 3\). Therefore, the domain of the function \(g\)is\(\left( { - 4,\;3} \right)\).

The function takes all values from \(\frac{1}{2}\)to\(4\). Therefore, the range of the function \(g\)is\(\left( {\frac{1}{2},\;4} \right)\).

Therefore, the domain and range of \(g\)are\(\left( { - 4,\;3} \right)\), and\(\left( {\frac{1}{2},\;4} \right)\), respectively.