Function notation, domain and range Problem set B One question at a time

\(g(x)=2x^2-7x+3\).

1. (a) Find \(g(4)\).

   • \(g(4)=2(16)-7(4)+3\)
   • \(g(4)=7\)

2. (b) Find \(g(-2)\).

   • \(g(-2)=2(4)+14+3\)
   • \(g(-2)=25\)

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

   • \(2x^2-7x+3=3 \;\Rightarrow\; 2x^2-7x=0\)
   • \(x(2x-7)=0\)
   • \(x=0\) or \(x=3.5\)

Write down the largest possible domain of each function.

1. (a) \(p(x)=\dfrac{9}{x+2}\)

   • The denominator cannot be 0: \(x+2\neq0\)
   • \(x\neq-2\)

2. (b) \(q(x)=\sqrt{3x-12}\)

   • The square root needs \(3x-12\geq0\)
   • \(x\geq4\)

3. (c) \(r(x)=\ln(x+4)\)

   • The logarithm needs \(x+4>0\)
   • \(x>-4\)

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

1. (a) \(p(x)=(x+1)^2-5\)

   • \((x+1)^2\geq0\), so \(p(x)\geq-5\)

2. (b) \(q(x)=-3x^2+12x-2\)

   • The graph is a hill. From the calculator: maximum at \((2,\,10)\)
   • \(y\leq10\)

3. (c) \(r(x)=5^x-2\)

   • \(5^x\) is always positive, so \(r(x)>-2\)

The graph shows \(y=g(x)\). The closed dot is at \((-1,\,5)\) and the open dot at \((5,\,5)\). Write down the domain and the range of g.

• Read left to right: \(-1\leq x<5\). The closed dot is included, the open dot is not.
• The lowest y-value is the minimum at \((2,\,-4)\). The highest is 5, reached at the closed dot.
• Range: \(-4\leq y\leq5\)

A taxi company charges \(F(k)=3.5+1.6k\) pounds for a journey of k kilometres.

1. (a) Find \(F(20)\).

   • \(F(20)=3.5+1.6(20)=35.5\)

2. (b) Find the length of a journey that costs £51.50.

   • \(3.5+1.6k=51.5 \;\Rightarrow\; 1.6k=48\)
   • \(k=30\)

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

   • \(F(0)=3.5\)
   • The starting charge is £3.50.

A parcel is dropped from a drone. Its height above the ground after t seconds is \(h(t)=125-5t^2\) metres. State a reasonable domain and range for the model.

• Time starts at 0. The parcel stops when \(h=0\).
• \(125-5t^2=0 \;\Rightarrow\; t^2=25 \;\Rightarrow\; t=5\)
• Domain: \(0\leq t\leq5\)
• The height falls from \(h(0)=125\) to \(h(5)=0\): range \(0\leq h\leq125\)

A school has 80 m of edging for a rectangular flower bed. The width is x metres, so the area is \(A(x)=x(40-x)\) square metres. State a reasonable domain and range for the model.

• A width must be positive, and the length \(40-x\) must be positive.
• Domain: \(0<x<40\)
• From the calculator: maximum at \((20,\,400)\)
• The area is always positive: range \(0<A\leq400\)

Find the largest possible domain of \(f(x)=\dfrac{\sqrt{x+5}}{x-2}\), and the range of \(g(x)=\dfrac{6}{x+1}\), where \(x\neq-1\).

• The square root needs \(x+5\geq0\), so \(x\geq-5\).
• The denominator cannot be 0, so \(x\neq2\).
• \(\dfrac{6}{x+1}\) is never 0, because the top is 6.
• Domain \(x\geq-5\), \(x\neq2\)\; range \(y\neq0\)

The function f is defined by \(f(x)=x^2-4x+1\) for \(0\leq x\leq6\). Find the range of f.

• From the calculator: minimum at \((2,\,-3)\)
• The end values are \(f(0)=1\) and \(f(6)=13\).
• The lowest output is \(-3\) and the highest is 13.
• \(-3\leq y\leq13\)