Question Statement

Find the graphical solution for the following pairs of functions:

(i) $f(x)=4-3x$ and $g(x)=-x+1$

(ii) $f(x)=2(2+x)$ and $g(x)=x^2+1$

(iii) $f(x)=5+3x$ and $g(x)=-x^2+5$

(iv) $f(x)=1$ and $g(x)=-2x^2+2x+5$

(v) $f(x)=2+3x+x^2$ and $g(x)=5+3x-2x^2$

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Background and Explanation

To find the graphical solution of two functions $f(x)$ and $g(x)$, we plot both equations on the same coordinate plane. The solution to the equation $f(x)=g(x)$ corresponds to the $x$-coordinates of the points where the two graphs intersect.

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Solution

Part (i)

We are looking for the intersection of two linear functions. First, we calculate coordinates to plot the lines.

Table for $f(x)=4-3x$:

Table for $g(x)=-x+1$:

By plotting these lines, we find the point where they cross.

Result: The point of intersection is $\left(\frac{3}{2},-\frac{1}{2}\right)$.

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Part (ii)

Here we find the intersection of a linear function $f(x)=4+2x$ and a quadratic function $g(x)=x^2+1$.

Table for $f(x)$:

Table for $g(x)$:

Result: The solutions (points of intersection) are $(-1,2)$ and $(3,10)$.

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Part (iii)

We compare the linear function $f(x)=5+3x$ and the downward-opening parabola $g(x)=-x^2+5$.

Table for $f(x)$:

Table for $g(x)$:

Result: The solutions are $(0,5)$ and $(-3,-4)$.

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Part (iv)

We find where the horizontal line $f(x)=1$ intersects the parabola $g(x)=-2x^2+2x+5$.

Table for $g(x)$:

Result: The solutions are $(-1,1)$ and $(2,1)$.

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Part (v)

We find the intersection of two quadratic functions: $f(x)=x^2+3x+2$ and $g(x)=-2x^2+3x+5$.

Combined Table of Values:

Result: The points of intersection are $(-1,0)$ and $(1,6)$.

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Key Formulas or Methods Used

• Point-Plotting Method: Calculating specific $(x,y)$ pairs to draw the shape of a function.
• Linear Functions: Equations of the form $y=mx+c$ which result in straight lines.
• Quadratic Functions: Equations of the form $y=a{x}^2+bx+c$ which result in parabolas.
• Graphical Intersection: The solution to $f(x)=g(x)$ is found at the coordinates $(x,y)$ where the graphs overlap.

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Summary of Steps

1. Create a table of values for both functions $f(x)$ and $g(x)$ by substituting various $x$ values.
2. Plot the resulting points on a Cartesian coordinate plane.
3. Connect the points to draw the line (for linear functions) or the curve (for quadratic functions).
4. Identify the points where the two graphs cross each other.
5. The $x$ and $y$ values of these intersection points represent the solutions.