Every solution shows the work, with the final answer in bold. Compare your reasoning, not only your result: for graphs, check that you drew the vertical lines, and for stories, check that your answer includes units.
Problem 1: Function from ordered pairs
List the inputs: 1, 2, 3, 4. No input is repeated, so each input has exactly one output.
Yes, it is a function.
Problem 2: Repeated input
The input 2 appears twice: it is paired with 5 and also with 9. One input has two outputs.
No, it is not a function.
Problem 3: Mapping diagram
Inputs 1, 2, 3, 4 each have exactly one arrow. The outputs 5, 5, 7, 9 repeat the 5, which is allowed.
Yes, it is a function.
Problem 4: Function from a table
The input \(x=2\) appears twice, with outputs 4 and 6.
No, \(y\) is not a function of \(x\).
Problem 5: Vertical line test: curve
Draw the vertical line \(x=1\). The curve satisfies \(1=y^{2}-3\), so \(y^{2}=4\) and \(y=2\) or \(y=-2\). The line meets the graph twice.
Not a function.
Problem 6: Vertical line test: V shape
Every vertical line meets the V-shaped graph in exactly one point.
It is a function.
Problem 7: Evaluate a linear function
\(f(4)=3(4)-5=12-5=7\)
\(f(-2)=3(-2)-5=-6-5=-11\)
\(f(0)=3(0)-5=-5\)
\(f(4)=7\), \(f(-2)=-11\), \(f(0)=-5\)
Problem 8: Evaluate a quadratic function
\(g(3)=9+6-8=7\)
\(g(-4)=(-4)^{2}+2(-4)-8=16-8-8=0\)
\(g(0)=0+0-8=-8\)
\(g(3)=7\), \(g(-4)=0\), \(g(0)=-8\)
Problem 9: Evaluate an absolute value function
\(h(1)=|1-4|+3=|-3|+3=3+3=6\)
\(h(4)=|0|+3=3\)
\(h(9)=|5|+3=8\)
\(h(1)=6\), \(h(4)=3\), \(h(9)=8\)
Problem 10: Negative and fractional inputs
\(k(-3)=-2(9)+5(-3)=-18-15=-33\). Square first: \((-3)^{2}=9\).
\(k(\frac{1}{2})=-2\cdot\frac{1}{4}+\frac{5}{2}=-\frac{1}{2}+\frac{5}{2}=2\)
\(k(-3)=-33\) and \(k(\frac{1}{2})=2\)
Problem 11: Find the input
\(2x+7=19\), so \(2x=12\) and \(x=6\). Check: \(2(6)+7=19\).
\(2x+7=-5\), so \(2x=-12\) and \(x=-6\). Check: \(2(-6)+7=-5\).
\(x=6\) and \(x=-6\)
Problem 12: Read a function table
Find the column with \(x=2\): the output is 0, so \(f(2)=0\).
The column \(x=-1\) has output 3, so \(f(-1)=3\).
Outputs equal to 3 appear for \(x=-1\) and \(x=3\).
\(f(2)=0\), \(f(-1)=3\), and \(x=-1\) or \(x=3\)
Problem 13: Read a function graph
At \(x=0\) the graph is at \(y=-3\). At \(x=4\) it is at \(y=5\). The graph touches the \(x\)-axis at \(x=-1\) and \(x=3\).
\(f(0)=-3\), \(f(4)=5\), and \(x=-1\) or \(x=3\)
Problem 14: Input with a variable
\(f(a+1)=3(a+1)-2=3a+3-2=3a+1\)
\(f(2a)=3(2a)-2=6a-2\)
\(f(a+1)=3a+1\) and \(f(2a)=6a-2\)
Problem 15: Domain and range of a segment
The graph runs from \(x=-2\) to \(x=4\), and both endpoints are closed. It runs from \(y=1\) to \(y=5\).
Domain: \(-2\le x\le 4\). Range: \(1\le y\le 5\).
Problem 16: Domain and range of a parabola
The graph extends left and right without end, so every real number is an input. The highest point is the vertex \((-1,3)\) and the graph goes down forever.
Domain: all real numbers. Range: \(y\le 3\).
Problem 17: Cupcake sale (discrete)
Only whole numbers of cupcakes make sense: the domain is \(\{1,2,3,4,5,6\}\). Then \(C(n)=3n\) gives \(3, 6, 9, 12, 15, 18\).
Domain: \(\{1,2,3,4,5,6\}\). Range: \(\{3,6,9,12,15,18\}\) dollars. Discrete.
Problem 18: Birthday relation
Each student has exactly one birthday, so every input (student) has one output (date): students to birthdays is a function.
Two students in the same class can share a birthday. Then one date would be paired with two students, so birthdays to students is not a function.
Problem 19: Plumber charges
The input 3 is hours and the output 115 is dollars. So a 3-hour job costs \(\$115\). Check: \(25(3)+40=115\).
\(C(2.5)=25(2.5)+40=62.5+40=102.5\)
A 3-hour job costs \(\$115\); \(C(2.5)=\$102.50\).
Problem 20: Height of a tossed ball
Check: \(h(2)=-16(4)+48(2)+4=-64+96+4=36\). It means that 2 seconds after the toss, the ball is 36 feet high.
\(h(1.5)=-16(2.25)+48(1.5)+4=-36+72+4=40\)
At 2 seconds the ball is 36 ft high; \(h(1.5)=40\) ft.
Problem 21: Filling a tank (continuous)
The tank is full when \(5t=40\), so \(t=8\) minutes. Time can be any value from 0 to 8, and the volume can be any value from 0 to 40.
Domain: \(0\le t\le 8\). Range: \(0\le V\le 40\). Continuous.
Problem 22: Taxi fare
\(F(6)=2.50+1.75(6)=2.50+10.50=13\), so a 6-mile ride costs \(\$13.00\).
Solve \(2.50+1.75m=20\): \(1.75m=17.50\), so \(m=10\).
\(F(6)=\$13.00\); a \(\$20.00\) ride is 10 miles.
Problem 23: Garden area
\(A(8)=8(12)=96\).
Test widths: \(A(5)=5(15)=75\) and \(A(15)=15(5)=75\), so \(w=5\) or \(w=15\).
Both width and length must be positive, so \(0 < w < 20\). The area is largest at \(w=10\): \(A(10)=100\). The area is always positive, so \(0 < A\le 100\).
\(A(8)=96\) ft\(^{2}\); \(w=5\) or \(15\) ft; domain \(0 < w < 20\); range \(0 < A\le 100\).
See every chapter on the Algebra 1 math page.