Function notation, domain and range Problem set A Worked solutions

\(f(x)=3x^2-5x+2\).

1. (a) Find \(f(3)\).

   • \(f(3)=3(9)-5(3)+2\)
   • \(f(3)=14\)

2. (b) Find \(f(-1)\).

   • \(f(-1)=3(1)+5+2\)
   • \(f(-1)=10\)

3. (c) Find the values of x for which \(f(x)=2\).

   • \(3x^2-5x+2=2 \;\Rightarrow\; 3x^2-5x=0\)
   • \(x(3x-5)=0\)
   • \(x=0\) or \(x=\frac{5}{3}=1.67\)

Write down the largest possible domain of each function.

1. (a) \(g(x)=\dfrac{7}{x-5}\)

   • The denominator cannot be 0: \(x-5\neq0\)
   • \(x\neq5\)

2. (b) \(h(x)=\sqrt{2x+6}\)

   • The square root needs \(2x+6\geq0\)
   • \(x\geq-3\)

3. (c) \(k(x)=\ln(x-1)\)

   • The logarithm needs \(x-1>0\)
   • \(x>1\)

Write down the range of each function, for all x in its domain.

1. (a) \(f(x)=(x-3)^2+4\)

   • \((x-3)^2\geq0\), so \(f(x)\geq4\)

2. (b) \(g(x)=-2x^2+8x+1\)

   • The graph is a hill. From the calculator: maximum at \((2,\,9)\)
   • \(y\leq9\)

3. (c) \(h(x)=2^x+3\)

   • \(2^x\) is always positive, so \(h(x)>3\)

The graph shows \(y=f(x)\). The closed dot is at \((-1,\,-3)\) and the open dot at \((4,\,2)\). Write down the domain and the range of f.

• Read left to right: \(-1\leq x<4\). The closed dot is included, the open dot is not.
• The lowest y-value is \(-3\) (the closed dot). The highest is the maximum at \((2,\,6)\).
• Range: \(-3\leq y\leq6\)

A print shop charges \(C(n)=18+2.4n\) pounds to print n posters.

1. (a) Find \(C(25)\).

   • \(C(25)=18+2.4(25)=78\)

2. (b) Find the number of posters that cost £90.

   • \(18+2.4n=90 \;\Rightarrow\; 2.4n=72\)
   • \(n=30\)

3. (c) Say what \(C(0)\) means in this situation.

   • \(C(0)=18\)
   • The fixed charge (set-up cost) is £18.

A ball is dropped from a platform. Its height above the ground after t seconds is \(h(t)=80-5t^2\) metres. State a reasonable domain and range for the model.

• Time starts at 0. The ball stops when \(h=0\).
• \(80-5t^2=0 \;\Rightarrow\; t^2=16 \;\Rightarrow\; t=4\)
• Domain: \(0\leq t\leq4\)
• The height falls from \(h(0)=80\) to \(h(4)=0\): range \(0\leq h\leq80\)

A farmer uses 60 m of fencing to make a rectangular paddock. The width is x metres, so the area is \(A(x)=x(30-x)\) square metres. State a reasonable domain and range for the model.

• A width must be positive, and the length \(30-x\) must be positive.
• Domain: \(0<x<30\)
• From the calculator: maximum at \((15,\,225)\)
• The area is always positive: range \(0<A\leq225\)

Find the largest possible domain of \(f(x)=\dfrac{\sqrt{x+3}}{x-1}\), and the range of \(g(x)=\dfrac{4}{x-2}\), where \(x\neq2\).

• The square root needs \(x+3\geq0\), so \(x\geq-3\).
• The denominator cannot be 0, so \(x\neq1\).
• \(\dfrac{4}{x-2}\) is never 0, because the top is 4.
• Domain \(x\geq-3\), \(x\neq1\)\; range \(y\neq0\)

The function f is defined by \(f(x)=x^2-6x+5\) for \(0\leq x\leq8\). Find the range of f.

• From the calculator: minimum at \((3,\,-4)\)
• The end values are \(f(0)=5\) and \(f(8)=21\).
• The lowest output is \(-4\) and the highest is 21.
• \(-4\leq y\leq21\)