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Wize High School Grade 12 Pre-Calculus Textbook > Combination of Functions

Solving Equations & Inequalities Graphically

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Solving Equations & Inequalities Graphically

Solving Equations Numerically through Guess & Check

Let f(x)=g(x)f(x)=g(x) be an equation where both f(x)f(x) and g(x)g(x) are continuous functions.

To solve for xx:
  1. Graph f(x)f(x)
  2. Graph g(x)g(x)
  3. Estimate the point of intersection (x,y)(x,y)
  4. Use xx from the point of intersection and evaluate f(x)−g(x)=0f(x)-g(x)=0 and determine how close the difference is to 0.
  5. If required, test again using another estimation of xx.
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Example 1

Solve x=sin⁡xx=\sin{x} for x.x.

Let f(x)=xLet g(x)=sin⁡x\begin{array}{rcl} \text{Let}~f(x)&=&x\\\\ \text{Let}~g(x)&=&\sin{x} \end{array}

Then,


An approximate value of xx can be 0.

f(x)−g(x)=0x−sin⁡x=0Let x=00−sin⁡0=00=0✓\begin{array}{rcl} f(x)-g(x)&=&0\\\\ x-\sin{x}&=&0&&\colorTwo{\footnotesize{\text{Let}~x=0}}\\\\ 0-\sin{0}&=&0\\\\ 0&=&0&&\color{red}\checkmark \end{array}

No more testing is required.


Therefore, the solution is x=0x=0.
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Solving Equations Using Graphing Technology

Let f(x)=g(x)f(x)=g(x) be an equation where both f(x)f(x) and g(x)g(x) are continuous functions.

To solve for xx:
  1. Graph f(x)f(x)and g(x)g(x):Y=\boxed{\text{Y=}} + Y1=f(x)\boxed{Y_1=f(x)} + Y2=g(x)\boxed{Y_2=g(x)} + GRAPH\boxed{\text{GRAPH}}
  2. Determine the point of intersection (x,y)(x,y): 2nd\boxed{2^{\text{nd}}} + TRACE\boxed{\text{TRACE}} + 5:intersect\boxed{5:\text{intersect}}
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Example 2

Solve (12)x≤x\Bigg(\dfrac{1}{2}\Bigg)^x\leq\sqrt{x}.

Step 1.

Y=\boxed{\text{Y=}} + Y1=(12)x\boxed{Y_1=\Bigg(\dfrac{1}{2}\Bigg)^x} + Y2=x\boxed{Y_2=\sqrt{x}} + GRAPH\boxed{\text{GRAPH}}



Step 2.

2nd\boxed{2^{\text{nd}}} + TRACE\boxed{\text{TRACE}} + 5:intersect\boxed{5:\text{intersect}} ≈\approx (0.5,0.707)(0.5,0.707)


Verify:

(12)0.5≤0.50.5≤0.5✓\begin{array}{rcl} \Bigg(\dfrac{1}{2}\Bigg)^{0.5}&\leq&\sqrt{0.5}\\\\ \sqrt{0.5}&\leq&\sqrt{0.5}&&\color{red}\checkmark \end{array}


Therefore, x≥0.5x\geq{}0.5 is the solution.

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Example: Solving Equations & Inequalities Graphically

Solve log⁡x≥x−1\log{x}\geq{}x-1 graphically.

Step 1.

Y=\boxed{\text{Y=}} + Y1=log⁡x\boxed{Y_1=\log{x}} + Y2=x−1\boxed{Y_2=x-1} + GRAPH\boxed{\text{GRAPH}}




Step 2.

2nd\boxed{2^{\text{nd}}} + TRACE\boxed{\text{TRACE}} + 5:intersect\boxed{5:\text{intersect}} ≈\approx (0.1371,−0.8629)(0.1371,-0.8629)

2nd\boxed{2^{\text{nd}}} + TRACE\boxed{\text{TRACE}} + 5:intersect\boxed{5:\text{intersect}} ≈\approx (1,0)(1,0)

Verify:

log⁡x≥x−1log⁡0.1371≥0.1371−1−0.8629≥−0.8639✓\begin{array}{rcl} \log{x}&\geq{}&x-1\\\\ \log{0.1371}&\geq{}&0.1371-1\\\\ -0.8629&\geq{}&-0.8639&&\color{red}\checkmark \end{array} log⁡x≥x−1log⁡1≥1−10≥0✓\begin{array}{rcl} \log{x}&\geq{}&x-1\\\\ \log{1}&\geq{}&1-1\\\\ 0&\geq{}&0&&\color{red}\checkmark \end{array}


Therefore, 0.1371≤x≤10.1371\leq{}x\leq{}1 is the solution.

Practice: Solving Equations & Inequalities Graphically

Solve 2x=x22^{x}=x^2 for xx using a numerical guess and check strategy.

Practice: Solving Equations & Inequalities Graphically

Suppose cos⁡x≥log⁡x\cos\sqrt{x}\geq{}\log{x}. Which of the following is correct?



Practice: Solving Equations & Inequalities Graphically

Determine the general solution to cos⁡(x)=cot⁡x\cos({x})=\cot{x}.



Extra Practice
Solving Equations & Inequalities Graphically
Practice: Solving Equations & Inequalities Graphically
Determine the general solution to sec⁡(x)=cot⁡x\sec({x})=\cot{x}.
Solving Equations & Inequalities Graphically
Practice: Solving Equations & Inequalities Graphically
Determine the general solution to sin⁡(x)=tan⁡x\sin({x})=\tan{x}.
Solving Equations & Inequalities Graphically
Practice: Solving Equations & Inequalities Graphically
Suppose sin⁡x≥3x−5\sin\sqrt{x}\geq{}3^{x-5}. Determine the solution.
Solving Equations & Inequalities Graphically
Practice: Solving Equations & Inequalities Graphically
Suppose (12)x<log⁡x\Bigg(\dfrac{1}{2}\Bigg)^x<\log{x}. Determine the solution to the inequality.
Solving Equations & Inequalities Graphically
Practice: Solving Equations & Inequalities Graphically
Solve log⁡x=x2−1\log{x}=x^2-1 for xx using a numerical guess and check strategy.
Solving Equations & Inequalities Graphically
Practice: Solving Equations & Inequalities Graphically
Solve 3x=−x3^{x}=\sqrt{-x} for xx using a numerical guess and check strategy.
Combination of Functions
Determine the solution to log⁡2x≤−(x−5)2+10\log_2{x}\leq{}-(x-5)^2+10
Solving Equations & Inequalities Graphically
Solve sin⁡3x<2x\sin{^3\sqrt{x}}<2^x. Round to the nearest whole number.