Wize High School Algebra II Textbook (Common Core) > 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)f(x)=g(x) be an equation where both f(x)f(x)f(x) and g(x)g(x)g(x) are continuous functions. 

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

Solve x=sin⁡xx=\sin{x}x=sinx for x.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}Let f(x)Let g(x)​==​xsinx​

Then,

  

An approximate value of xxx 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}f(x)−g(x)x−sinx0−sin00​====​0000​​Let x=0✓​

No more testing is required. 


Therefore, the solution is x=0x=0x=0.

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Solving Equations Using Graphing Technology
Let f(x)=g(x)f(x)=g(x)f(x)=g(x) be an equation where both f(x)f(x)f(x) and g(x)g(x)g(x) are continuous functions. 

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

Solve (12)x≤x\Bigg(\dfrac{1}{2}\Bigg)^x\leq\sqrt{x}(21​)x≤x​. 

Step 1. 

Y=\boxed{\text{Y=}}Y=​ + Y1=(12)x\boxed{Y_1=\Bigg(\dfrac{1}{2}\Bigg)^x}Y1​=(21​)x​ + Y2=x\boxed{Y_2=\sqrt{x}}Y2​=x​​ + GRAPH\boxed{\text{GRAPH}}GRAPH​

Step 2.

2nd\boxed{2^{\text{nd}}}2nd​ + TRACE\boxed{\text{TRACE}}TRACE​ + 5:intersect\boxed{5:\text{intersect}}5:intersect​ ≈\approx≈ (0.5,0.707)(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}(21​)0.50.5​​≤≤​0.5​0.5​​​✓​

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

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Example: Solving Equations & Inequalities Graphically
Solve log⁡x≥x−1\log{x}\geq{}x-1logx≥x−1 graphically. 

Step 1. 

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




Step 2.

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

2nd\boxed{2^{\text{nd}}}2nd​ + TRACE\boxed{\text{TRACE}}TRACE​ + 5:intersect\boxed{5:\text{intersect}}5:intersect​ ≈\approx≈ (1,0)(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}logxlog0.1371−0.8629​≥≥≥​x−10.1371−1−0.8639​​✓​    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}logxlog10​≥≥≥​x−11−10​​✓​


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

Practice: Solving Equations & Inequalities Graphically
Solve 2x=x22^{x}=x^22x=x2 for xxx using a numerical guess and check strategy.

Smaller value of

x

(to the nearest hundredth)

Larger value of

x

I don't know

Practice: Solving Equations & Inequalities Graphically
Suppose cos⁡x≥log⁡x\cos\sqrt{x}\geq{}\log{x}cosx​≥logx. Which of the following is correct? 

Practice: Solving Equations & Inequalities Graphically
Determine the general solution to cos⁡(x)=cot⁡x\cos({x})=\cot{x}cos(x)=cotx. 

A) 
−1.57±2πn, n∈W-1.57\pm2\pi{}n,~n\in\mathbb{W}−1.57±2πn, n∈W

1.57±2πn, n∈W1.57\pm2\pi{}n,~n\in\mathbb{W}1.57±2πn, n∈W

B) 
−1.57±πn, n∈W-1.57\pm\pi{}n,~n\in\mathbb{W}−1.57±πn, n∈W

1.57±πn, n∈W1.57\pm\pi{}n,~n\in\mathbb{W}1.57±πn, n∈W

C)
−1.57±πn, n∈W-1.57\pm\pi{}n,~n\in\mathbb{W}−1.57±πn, n∈W

D)

1.57±πn, n∈W1.57\pm\pi{}n,~n\in\mathbb{W}1.57±πn, n∈W

I don't know

Extra Practice

Solving Equations & Inequalities Graphically

Practice: Solving Equations & Inequalities Graphically
Determine the general solution to sec⁡(x)=cot⁡x\sec({x})=\cot{x}sec(x)=cotx. 

Solving Equations & Inequalities Graphically

Practice: Solving Equations & Inequalities Graphically
Determine the general solution to sin⁡(x)=tan⁡x\sin({x})=\tan{x}sin(x)=tanx. 

Solving Equations & Inequalities Graphically

Practice: Solving Equations & Inequalities Graphically
Suppose sin⁡x≥3x−5\sin\sqrt{x}\geq{}3^{x-5}sinx​≥3x−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}(21​)x<logx. Determine the solution to the inequality.  

Solving Equations & Inequalities Graphically

Practice: Solving Equations & Inequalities Graphically
Solve log⁡x=x2−1\log{x}=x^2-1logx=x2−1 for xxx using a numerical guess and check strategy.

Solving Equations & Inequalities Graphically

Practice: Solving Equations & Inequalities Graphically
Solve 3x=−x3^{x}=\sqrt{-x}3x=−x​ for xxx 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+10log2​x≤−(x−5)2+10

Solving Equations & Inequalities Graphically

Solve sin⁡3x<2x\sin{^3\sqrt{x}}<2^xsin3x​<2x. Round to the nearest whole number.