Problem
1. Explain why g: x2 - 2x - y - 3 = 0 is a function, and write it using function notation.
2. Evaluate the function for the arguments { 5, t, 2x + 1 }.
3. Graph y = g(x) and use the vertical line test to verify it is a function.
4. State the domain and range of the function g.
Solution
1. Explain why g: x2 - 2x - y - 3 = 0 is a function, and write it using function notation.
Begin by solving for y in relation g: x2 - 2x - y - 3 = 0. The result is y = x2 - 2x - 3.
y = x2 - 2x - 3 is a function since any value substituted into x yields exactly one value as a result. y = x2 - 2x - 3 can be written using function notation as g(x) = x2 - 2x - 3.
2. Evaluate the function for the arguments { 4, t, 2x + 1 }.
g(x) = x2 - 2x - 3 Let x = t.
g(t) = ( t )2 - 2( t ) - 3
g(t) = t2 - 2t - 3
g(x) = x2 - 2x - 3 Let x be replaced by 2x + 1.
g(2x + 1) = ( 2x + 1 )2 - 2( 2x + 1 ) - 3
f(2x + 1) = 4x2 + 4x + 1 - 4x - 2 - 3
f(2x + 1) = 4x2 - 4
3. Graph y = g(x) and use the vertical line test to verify it is a function.
Image onlyInstructions text as in global.js
The vertical line test: If a vertical line can be drawn anywhere in the coordinate plane so that it intersects the graph of a relation at more than one location, then the relation is NOT a function.
In this example, a vertical line always intersects the graph of the relation at exactly one location (see the illustration below). As a result, the vertical line test indicates that the relation g (x2 - 2x - y - 3 = 0) is a function.
4. State the domain and range of the function g.
Since any real number substituted into the function g(x) results in another real number, the domain of function g is the real numbers.
Domain of g: { x | x ∈ R }
Graphically the domain can be "viewed" by imagining a vertical line moving right or left over the graph of the function (see the illustration above). Where the vertical line intersects the graph of the function, an x-value which is part of the domain of the function occurs. Where the vertical line does NOT intersect the graph of the function, an x-value occurs (based on the horizontal position of the line) which is NOT part of the domain of the function.
In this example, no matter where a vertical line is drawn it intersects the graph of the function. This indicates that the domain of the function is the Real Numbers.
Range of g: { y | y ≥ 4 }
Graphically the range can be "viewed" by imagining a horizontal line moving up or down over the graph of the function (see the illustration below). Where the horizontal line intersects the graph of the function, a y-value which is part of the range of the function occurs. Where the horizontal line does NOT intersect the graph of the function, a y-value occurs (based on the vertical position of the line) which is NOT part of the range of the function.
In this example, only when a horizontal line is drawn with a y-value greater than or equal to -4 does it intersect the graph of the function.
